## Adjustable subwavelength localization in a hybrid plasmonic waveguide |

Optics Express, Vol. 21, Issue 6, pp. 7427-7438 (2013)

http://dx.doi.org/10.1364/OE.21.007427

Acrobat PDF (2306 KB)

### Abstract

The hybrid plasmonic waveguide consists of a high-permittivity dielectric nanofiber embedded in a low-permittivity dielectric near a metal surface. This architecture is considered as one of the most perspective candidates for long-range subwavelength guiding. We present qualitative analysis and numerical results which reveal advantages of the special waveguide design when dielectric constant of the cylinder is greater than the absolute value of the dielectric constant of the metal. In this case the arbitrary subwavelength mode size can be achieved by controlling the gap width. Our qualitative analysis is based on consideration of sandwich-like conductor-gap-dielectric system. The numerical solution is obtained by expansion of the hybrid plasmonic mode over single cylinder modes and the surface plasmon-polariton modes of the metal screen and matching the boundary conditions.

© 2013 OSA

## 1. Introduction

2. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Phot. **4**, 83–91 (2010) [CrossRef] .

3. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “Hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Phot. **2**(8), 496–500 (2008) [CrossRef] .

4. V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. **2**, 331 (2011) [CrossRef] .

*tg*=

*ε*″

_{m}/

*ε*′

_{m}at the operating frequency and the specific spatial structure of the guiding mode with field confinement within non dissipative gap region.

5. I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express **18**(1), 348–363 (2010) [CrossRef] [PubMed] .

6. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A. **13**(3), 483–493 (1996) [CrossRef] .

7. R. Borghi, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface,” J. Opt. Soc. Am. A. **14**(7), 1500–1504 (1997) [CrossRef] .

## 2. Qualitative description

*d*and permittivity

*ε*is placed above a metal screen of permittivity

_{d}*ε*. The width of the gap between the cylinder and the metal screen is

_{m}*h*, see Fig. 1(a). Let us choose the Cartesian reference system as it is shown in Fig. 1:

*z*-axis is directed along the waveguide, whereas

*x*-axis is directed normally to the metal screen. We consider a plasmon-polariton mode of frequency

*ω*and the propagation constant

*β*propagating along

*z*-axis. Thus, all electromagnetic field components depend on time and

*z*-coordinate as exp[

*iβz*−

*iωt*]. We assume, that responses of both dielectric and metal on electromagnetic field are described by dielectric constants, which are

*ε*

_{m}and

*ε*

_{d}respectively. Generally, the outer medium may be not vacuum, but some dielectric medium having dielectric constant being equal to

*ε*

_{g}. All the materials are assumed to be nonmagnetic. To describe the mode confinement, it is convenient to introduce effective refractive index

*n*

_{eff}, which is defined as

*n*

_{eff}=

*β*/

*k*, where

*k*=

*ω*/

*c*in the wavenumber in vacuum. The effective index determines the field penetration depth into the material with permittivity

*ε*as

*n*

_{eff}is the stronger degree of confinement.

3. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “Hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Phot. **2**(8), 496–500 (2008) [CrossRef] .

5. I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express **18**(1), 348–363 (2010) [CrossRef] [PubMed] .

*n*

_{CGD}is greater than the refractive index of the dielectric

5. I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express **18**(1), 348–363 (2010) [CrossRef] [PubMed] .

3. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “Hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Phot. **2**(8), 496–500 (2008) [CrossRef] .

*d*otherwise HPP-mode should be considered as a result of hybridization of surface plasmon polariton modes and the modes of the single dielectric cylinder.

**2**(8), 496–500 (2008) [CrossRef] .

**18**(1), 348–363 (2010) [CrossRef] [PubMed] .

*ε*

_{m}| <

*ε*

_{d}. For the case, the effective refractive index

*n*

_{CGD}is proportional to inverse width of the gap,

*n*

_{CGD}∝ 1/

*kh*, when the width

*h*is small enough. To use the same effect for the hybrid waveguide, the cylinder diameter should sufficiently exceed some critical value

*d*

^{*}, which is determined by the condition that the transversal size of the plain part of the gap is comparable with the mode penetration depth into the dielectric

*d*greater enough than

*d*

^{*}the guiding mode can approach the strongly confined mode of the sandwich like system even if the diameter of the cylinder is much less than free space wavelength.

**18**(1), 348–363 (2010) [CrossRef] [PubMed] .

*i*=

*m*,

*g*,

*d*and

*n*

_{CGD}is the effective index of the mode. In particular, 1/

*κ*

_{d}and 1/

*κ*

_{m}are the penetration depths into the dielectric and the metal correspondingly. It is known that such plane three-layer waveguide supports the propagation of the bound eigen mode only if the width of the intermediate layer is less than some cut-off value

*h*which is determined by the permittivities at given frequency

_{c}*n*

_{CGD}on the gap thickness

*h*for the cases of low and high index dielectric, see Fig. 2. For relatively low refractive index of dielectric,

*ε*

_{d}< |

*ε*

_{m}|, there exists surface plasmon-polariton mode when the gap is absent,

*h*= 0. It has effective index

**18**(1), 348–363 (2010) [CrossRef] [PubMed] .

**2**(8), 496–500 (2008) [CrossRef] .

*n*

_{CGD}unlimitedly diverges as the gap thickness tends to zero,

9. I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metaldi-electric multilayers,” Phys. Rev. B **75**(24), 241402 (2007) [CrossRef] .

*h*<

*h*

_{c}from the metal plane. When such metal is involved the effective index of the HPP mode can be significantly greater than effective index of electromagnetic field in bulk material of the cylinder even for very small diameters of the cylinder. Note, that to calculate group velocity

*v*and chromatic dispersion for the mode using the formula (4), one should know the dispersion laws for permittivities

_{g}*ε*

_{m}and

*ε*

_{d}. For thin gap

*h*≪

*h*, the group velocity scales as

_{c}*v*/

_{g}*c*∝

*h*/

*λ*. Thus divergence of the CGD-mode effective index with the gap width decreasing leads to strong reduction of the group velocity.

**2**(8), 496–500 (2008) [CrossRef] .

*ε*

_{g}≪ |

*ε*

_{m}|. For the case where

*ε*″

_{m}is the imaginary part of the metal permittivity. It follows from Eq. (5), that the localization radius is of the order of

*h*|

*ε*| in the limit

_{m}*h*≪

*h*. Note, that our approach allows to squeeze the mode at arbitrary frequency into any subwavelength scale simply by tuning the gap width in accordance with (4). Hence, our waveguide design breaks connection between mode localization and the carrying frequency of the mode. In particular, the approach may be interesting for design waveguides at THz frequencies [10

_{c}10. A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep Subwavelength Terahertz Waveguides Using Gap Magnetic Plasmon”, Phys. Rev. Let. **102**, 043904 (2009) [CrossRef] .

11. S. H. Nam, A. J. Taylor, and A. Efimov, “Subwavelength hybrid terahertz waveguides,” Opt. Express. **17**(25), 22890–22897 (2009) [CrossRef] .

*ℓ*∼

*h*|

*ε*

_{m}|/|

*tg*| i.e. reduces with the mode size reduction. To keep the propagation length acceptable for practical implementation at fixed degree of localization one should minimize loss tangent

*tg*. Thus, a prospecting like [12

12. P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. **4**, 795–808 (2010) [CrossRef] .

## 3. Semi-analytical description and numerical results

*z*-components of the electric and the magnetic fields,

*E*and

_{z}*B*[13]. These fields satisfy the following two-dimensional Helmholtz differential equation inside the homogeneous areas where permittivity is constant: where

_{z}*k*=

*ω*/

*c*is the free space wavenumber. The boundary conditions on the both interfaces are continuity of components

*E*,

_{z}*H*,

_{z}*εE*and

_{ξ}*H*, where

_{ξ}*ξ*-component of a vector is its normal component.

15. W. Zakowicz, “Two coupled dielectric cylindrical waveguides,” J. Opt. Soc. Am. A **14**(3), 580–587 (1997) [CrossRef] .

*d*for a range of the gap widths

*h*in the case of telecommunication wavelength when

*ε*

_{g}<

*ε*

_{d}< |

*ε*

_{m}|. These dispersion curves are obtained from our numerical procedure and show a good agreement with the results obtained in [3

**2**(8), 496–500 (2008) [CrossRef] .

*ε*

_{d}< |

*ε*

_{m}|, the parameters of the waveguide are taken accordingly to experimental work [16

16. R. F. Oulton, V. J. Sorger, T. Zentgraf, Ren-Min Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**, 629–632 (2009) [CrossRef] [PubMed] .

*ε*

_{d}> |

*ε*

_{m}|. Parameters of these two plots differ only for metal permittivity

*ε*

_{m}, the value

*ε*

_{m}= −4 is chosen for Fig. 4(b). Here, we do not concretize the material of the metal screen, our goal is just to demonstrate the qualitative difference of the guiding mode properties for the case (3).

*d*is decreased, the HPP-mode loses confinement along the metal and eventually (at

*d*= 0) becomes a surface plasmon-polariton mode of the flat metal-vacuum interface. Herewith the effective index of the HPP-mode monotonically decreases to that of this SPP-mode. Thus all dispersion curves have the same asymptotic

*d*. Two different behavior are possible at the opposite limit of large diameter. As the diameter

*d*→ ∞, the HPP-mode can asymptotically tend either fundamental single fiber mode or the fundamental mode of the planar three-layer system, the choice depends on the gap width

*h*. If the gap thickness

*h*is below than

*h*

_{c}(Eq. (2)) the HPP-mode approaches the CGD-mode with the diameter increasing. In the case the crossover between the asymptotics occurs at

*d*

^{*}(black arrows at Fig. 4(b)) which is determined as

*h*>

*h*

_{c}the HPP-mode becomes the cylinder-like in the limit of the large diameter. In the case the critical diameter

*d*

_{0}corresponding to the transition between small-diameter and large-diameter asymptotics is defined by the equation

*n*

_{SF}(

*d*

_{0}) =

*n*

_{mg}, where

*n*

_{SF}(

*d*) is the diameter dependence of the effective index of the single fiber fundamental mode. If the condition

*γ*= 0.5772... is Euler-Mascheroni constant.

*n*

_{CGD}of the CGD-mode not to be significantly above that of bulk plane wave in the fiber medium,

*ε*

_{m}| >

*ε*

_{d}. Just the case is realised at Fig. 3 and Fig. 4(a). Then the field penetration depth into the upper dielectric is quite large as well as the HPP-to-CGD crossover diameter

*d*

^{*}>

*λ*, so the CGD-mode does not provide the strong confinement. Therefore there are no advantages of CGD-like limit in the case from the view of HPP-mode confinement. For a given frequency and gap width the choice with the strongest coupling of the fiber mode and the surface plasmon polariton mode, corresponding to

*d*=

*d*

_{0}, provides the strongest localization of the field within nanogap due to the great contrast of permittivities [3

**2**(8), 496–500 (2008) [CrossRef] .

*h*= 2

*nm*. At the same time significant part of energy is transferred inside the fiber, thus the waveguide mode confinement is achieved largely due to the boundedness of the high-permittivity dielectric part of the waveguide. Once the diameter of the fiber is optimum and the gap width is small enough the advantages of the hybrid architecture are used completely: cross section size of the system can be much less than the wavelength and mode confinement is much stronger than for uncoupled single fiber or flat metal-dielectric interface. To achieve further increase of the HPP-mode confinement the fiber with higher dielectric constant should be used.

*ε*| <

_{m}*ε*that corresponds to the Fig. 4(b). Then the CGD-mode has strong confinement so the crossover diameter can be decreased to deep subwavelength scale,

_{d}*d*

^{*}≪

*λ*, by tuning the gap width. Therefore the attractive CGD-like asymptotic can be achieved by HPP-mode with very small diameter of cylinder providing the wished structure of the mode with the strong transversal localization in two dimensions within the gap region and exponential decaying into the cylinder. The example of the spatial distribution of the time-average Poynting vector is represented in Fig. 5(b). Note that in the case the top part of the fiber cross section is at distances much larger than 1/

*κ*

_{d}from the gap and its particular shape does not play role any more. As it was mentioned above the propagation distance decreases with the gap width decreasing, see Eq. (5). We present the results of simulation performed by using the COMSOL commercial software package in Fig. 6. The ultra-small mode confinement leads to storage of the large portion of the electromagnetic energy within dissipative metal region. Thus the stronger localization corresponds to the shorter HPP-mode’s propagation length.

## 4. Conclusion

*ε*| <

_{m}*ε*are confirmed by both qualitative analysis within planar three-layer model and rigorous semi-analytical method describing the HPP-mode propagation in general. The propagation distance of hybrid mode reduces with the mode size reduction. To achieve long-range propagation at fixed degree of localization one should minimize loss tangent

_{d}*tg*= |

*ε*″

*/*

_{m}*ε*′

*| of the metal. It should be noted that simultaneous satisfying of both conditions |*

_{m}*ε*| <

_{m}*ε*and

_{d}*tg*≪ 1 at optical and near infrared frequencies is a challenging task. Thus implementation of the waveguide loss compensation techniques would be required to use such hybrid waveguide as a component of the miniaturized photonic circuits. Another potential application of our waveguide design lies in study field of the resonant plasmons. The resonance condition for a surface plasmon-polariton at a planar metal-dielectric interface is the fine-tuning of the permittivities, −(

*ε*+

_{m}*ε*) ≪

_{d}*ε*[17

_{d}17. I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. B **70**, 155416 (2004) [CrossRef] .

*h*≪

*λ*. Thus for the conductor-gap-dielectric structure the resonance of plasmonic mode can be attained by gap width decreasing at any frequency as long as the condition |

*ε*| <

_{m}*ε*is valid.

_{d}## A. Numerical method

*z*-axis (Fig. 7). Supposing the structure of the fundamental hybrid mode to be symmetric with respect to

*x*-axis we can describe the longitudinal component of the electric field as

*z*-components of the magnetic field as

*D*=

*d*/2 +

*h*.

18. R. Borghi, F. Frezza, M. Santarsiero, and G. Schettini, “Angular spectrum of modified cylindrical wave-functions,” Int. J. Infrared Millim. Waves **20**(10), 1795–1801 (1999) [CrossRef] .

*η*is tangent to the interface coordinate in the transversal plane.

*E*,

_{z}*B*,

_{z}*εE*and

_{x}*εB*lead to the first system of linear homogeneous equations (SLE) on coefficients

_{x}*E*,

_{z}*B*,

_{z}*εE*and

_{r}*εB*on the cylindrical surface produce the second SLE on amplitudes

_{r}## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Phot. |

3. | R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “Hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Phot. |

4. | V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. |

5. | I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express |

6. | R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A. |

7. | R. Borghi, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface,” J. Opt. Soc. Am. A. |

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

9. | I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metaldi-electric multilayers,” Phys. Rev. B |

10. | A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep Subwavelength Terahertz Waveguides Using Gap Magnetic Plasmon”, Phys. Rev. Let. |

11. | S. H. Nam, A. J. Taylor, and A. Efimov, “Subwavelength hybrid terahertz waveguides,” Opt. Express. |

12. | P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. |

13. | D. Marcuse, |

14. | A. G. Bulushev, E. M. Dianov, and O. G. Okhotnikov, “Propagation of the radiation in two identical coupled waveguides,” Quantum Electron. |

15. | W. Zakowicz, “Two coupled dielectric cylindrical waveguides,” J. Opt. Soc. Am. A |

16. | R. F. Oulton, V. J. Sorger, T. Zentgraf, Ren-Min Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature |

17. | I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. B |

18. | R. Borghi, F. Frezza, M. Santarsiero, and G. Schettini, “Angular spectrum of modified cylindrical wave-functions,” Int. J. Infrared Millim. Waves |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 8, 2013

Revised Manuscript: March 2, 2013

Manuscript Accepted: March 4, 2013

Published: March 18, 2013

**Citation**

S. Belan, S. Vergeles, and P. Vorobev, "Adjustable subwavelength localization in a hybrid plasmonic waveguide," Opt. Express **21**, 7427-7438 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-7427

Sort: Year | Journal | Reset

### References

- M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
- D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Phot.4, 83–91 (2010). [CrossRef]
- R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “Hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Phot.2(8), 496–500 (2008). [CrossRef]
- V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun.2, 331 (2011). [CrossRef]
- I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express18(1), 348–363 (2010). [CrossRef] [PubMed]
- R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A.13(3), 483–493 (1996). [CrossRef]
- R. Borghi, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a dielectric circular cylinder parallel to a general reflecting flat surface,” J. Opt. Soc. Am. A.14(7), 1500–1504 (1997). [CrossRef]
- J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett.82(8), 1158–1160 (1997).
- I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metaldi-electric multilayers,” Phys. Rev. B75(24), 241402 (2007). [CrossRef]
- A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep Subwavelength Terahertz Waveguides Using Gap Magnetic Plasmon”, Phys. Rev. Let.102, 043904 (2009). [CrossRef]
- S. H. Nam, A. J. Taylor, and A. Efimov, “Subwavelength hybrid terahertz waveguides,” Opt. Express.17(25), 22890–22897 (2009). [CrossRef]
- P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev.4, 795–808 (2010). [CrossRef]
- D. Marcuse, Light Transmission Optics (New York: Van Nostrand Reinhold, 1972).
- A. G. Bulushev, E. M. Dianov, and O. G. Okhotnikov, “Propagation of the radiation in two identical coupled waveguides,” Quantum Electron.15, 1433–1441 (1988).
- W. Zakowicz, “Two coupled dielectric cylindrical waveguides,” J. Opt. Soc. Am. A14(3), 580–587 (1997). [CrossRef]
- R. F. Oulton, V. J. Sorger, T. Zentgraf, Ren-Min Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature461, 629–632 (2009). [CrossRef] [PubMed]
- I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. B70, 155416 (2004). [CrossRef]
- R. Borghi, F. Frezza, M. Santarsiero, and G. Schettini, “Angular spectrum of modified cylindrical wave-functions,” Int. J. Infrared Millim. Waves20(10), 1795–1801 (1999). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.