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Optics Express

  • Editor: J. H. Eberly
  • Vol. 9, Iss. 10 — Nov. 5, 2001
  • pp: 490–497
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Cylindrical vector beam focusing through a dielectric interface

D. P. Biss and T. G. Brown  »View Author Affiliations


Optics Express, Vol. 9, Issue 10, pp. 490-497 (2001)
http://dx.doi.org/10.1364/OE.9.000490


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Abstract

Cylindrical vector beams have been proposed and demonstrated for applications ranging from microscopy to high energy physics. In this paper, we analyze the three-dimensional field distributions of radial and azimuthal beams focused near a dielectric interface. We give particular attention to the classic problem of high numerical aperture focusing from an immersion lens to a glass-air interface and find that the use of radially and azimuthally polarized illumination for this type of imaging provides an impressive lateral confinement of the fields over a wide range of interface positions.

© Optical Society of America

1. Introduction

Cylindrical-vector beams are solutions of Maxwell’s equations which obey cylindrical symmetry both in field-amplitude and polarization. Such solutions have been the topic of numerous recent theoretical and experimental investigations [5

5. Lars Egil Helseth, “Roles of polarization, phase, and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

23

23. T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, “Polarization spectroscopy of single fluorescent molecules,” J. Phys. Chem. B 103, 6839–6850 (1999) [CrossRef]

]. It has been found, for example, that such beams may be produced by active or passive means [13

13. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234–2239 (1990). [CrossRef] [PubMed]

17

17. E. G. Churin, J. Hossfeld, and T. Tschudi, “Polarization configurations with singular point formed by computer generated holograms,” Opt. Commun. 99, 13–17 (1993). [CrossRef]

]. The use of an interferometric polarization converter has recently allowed the application of focused cylindrical-vector beams to several different types of microscopy [14

14. K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE3919 (2000). [CrossRef]

].

Meanwhile, the propagation and focusing properties of these beams remain of continued interest. Three recent papers [14

14. K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE3919 (2000). [CrossRef]

,18

18. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Exp. 7, 77–87 (2000). [CrossRef]

,19

19. S. Quabis, R. Dorn, M. Eberler, O. G. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

], have analyzed the high numerical-aperture focusing properties of the beams in free space, and have shown that a highly inhomogeneous distribution of electric field directions exist near focus and that, in the case of a radially polarized beam, the non-propagating longitudinal component to the electric field may achieve a strength several times that of the propagating component.

Fields focused onto an interface have been previously investigated [2

2. Hao Ling and Shung-Wu Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984). [CrossRef]

4

4. S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997). [CrossRef]

], these papers have only considered linearly polarized light. Recently Helseth has considered interfacial focusing with beams of arbitrary polarization and gave one example of a radially polarized beam [5

5. Lars Egil Helseth, “Roles of polarization, phase, and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

]. The strong longitudinal component created by radially polarized beams naturally raises the question of the distribution of fields at various interface positions. Recent investigations of optical focusing inside a dielectric have demonstrated that focusing a source of illumination over a cone of light which extends beyond the critical angle produces astigmatic effects due to the Goos-Hänchen phase shift experienced by plane waves undergoing total internal reflection.

In this paper, we analyze the through focus three-dimensional field distributions of a cylindrical vector beam incident on a dielectric interface, with special attention to using such beams for high numerical aperture illumination from inside a dielectric. This is of considerable interest because, as we will show, a cylindrical vector beam, by its very nature, possesses a plane wave spectrum which is either entirely s-like or entirely p-like. There is therefore no astigmatism and the lateral confinement of the fields near focus is impressive over a wide range of interface positions.

2. Theory

To find the reflected and transmitted fields of the focused field, an initial focused field is found using the theory described by Richards and Wolf [1

1. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959). [CrossRef]

] and applied to cylindrical vector beams by Youngworth and Brown [18

18. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Exp. 7, 77–87 (2000). [CrossRef]

].

Fig. 1. Aplanatic lens focusing onto an interface

An initial field, represented by an apodization amplitude function, o (θ), is refracted by an aplanatic lens and focused near the interface as shown in figure 1. The forms of the azimuthally and radially polarized beams are found in the same fashion as Török et al [3

3. P. Török, P. Varga, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995). [CrossRef]

] and Helseth [5

5. Lars Egil Helseth, “Roles of polarization, phase, and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

].

2.1 Azimuthally Polarized Light

For azimuthally polarized light the procedure is similar to that of Helseth [5

5. Lars Egil Helseth, “Roles of polarization, phase, and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

], the only differences being that the incoming light is completely s polarized. The initial field after refraction by the aplanatic lens is,

Ef(θ,ϕ)=o(θ)cos12θ[sinϕcosϕ0].
(1)

This field is propagated to the interface by the usual method of the angluar spectrum representation. Reflected and transmitted fields are assumed and Maxwell’s boundary conditions are applied to these fields. The results are,

Ef(ρ,z)=0θmaxo(θ)Efcos12θsinθdθ,
(2)
Ef(ρ,z)=0θmaxo(θ)Efcos12θsinθdθ,
(3)
Et(ρ,z)=0θmaxo(θ)Efcos12θsinθdθ,
(4)

with

Ef=eik1zcosθ[0J1(ρk1sinθ)0],
(5)
Er=rseik1zcosθei2k1z0cosθ[0J1(ρk1sinθ)0],
(6)
Et=tseik1z0cosθeik21(k1k2sinθ)2(zz0)[0J1(ρk1sinθ)0].
(7)

The integration of the azimuthal angle can now be performed numerically. Care has to be taken when integrating if the entering angles exceed the critical angle of the system due to the fact that complex poles exist in the integrand for azimuthal angles at the critical angle.

2.2 Radially Polarized Light

The case of radially polarized light is similar and this polarization has been considered by Helseth [5

5. Lars Egil Helseth, “Roles of polarization, phase, and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

]. The result has the same integral form as azimuthal light, but with different Ef , Er , and Et ,

Ef=eik1zcosθ[icosθJ1(ρk1sinθ)0sinθJ0(ρk1sinθ)],
(8)
Er=rpeik1zcosθei2k1zocosθ[icosθJ1(ρk1sinθ)0sinθJ0(ρk1sinθ)],
(9)
Et=tpeik1z0cosθeik21(k1k2sinθ)2(zz0)[i1(k1k2sinθ)2J1(ρk1sinθ)0k1k2sinθJ0(ρk1sinθ)].
(10)

3. Results and Discussion

So far the solutions for the reflected and transmitted fields have been left in a general form. No apodization function has been chosen. Here we will choose an apodization function originally derived by Jordan and Hall [7

7. R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427–429 (1994). [CrossRef] [PubMed]

], which describes the Bessel-Gauss class of fields. The lowest order field is given by,

o(θ)=Eoexp(β2sin2θsin2θmax)J1(2βsinθsinθmax).
(11)

Here θ max is the maximum entering angle of the beam, and β is a ratio of the pupil radius to the entering beam radius. In these examples β=1, which indicates that the pupil fill is equal to 1. With this apodization function, eqs. (2)(10) must be evaluated numerically. The follow results were obtained using a Gaussian quadrature integration method.

Figures 2a and 2b show the radial and longitudinal field intensities at an interface for two sets of parameters. Figure 2a considers an oil immersion lens focusing into air, and figure 2b examines focusing in air onto a silicon (n=3.55) substrate, such as might be required in semiconductor inspection.

Fig. 2. a. Focusing of radial component with incident medium (left side) n=1.518 and exiting medium (right side) n=1. The NA for the system is 1.4. The white line represents the interface of the system. The color scale is a log base 10 scale in arbitrary units. Axes are in units of wavelength. The picture on the left is the radial component. The picture on the right is the longitudinal component.
Fig. 2. b. Focusing of radial component with incident medium (left side) n=1 and exiting medium (right side) n=3.55. The NA for the system is .85. The white line represents the interface of the system. The color scale is a log base 10 scale in arbitrary units. Axes are in the units of wavelength. The picture on the left is the radial component. The picture on the right is the longitudinal component.

Here we see that the radial field component is continuous across the interface, while the longitudinal component is discontinuous. This is expected according to Maxwell’s boundary conditions.

The fields for azimuthally polarized light can be obtained using a similar procedure. In this case we find only one vector component in the azimuthal direction and the field is everywhere s polarized. In Figures 3a and 3b we see two examples of azimuthally polarized light focused onto an interface. Figure 3a considers an oil immersion lens focusing in air, and figure 3b considers focusing from air onto a silicon substrate. We see again that the on axis null is preserved through the focus. Also since the field is transverse to the plane of the interface, there is no field discontinuity as the beam passes through the interface.

Fig. 3. a. Azimuthally polarized light focused onto an interface with NA=1.4 and entering medium n=1.518, and exiting medium n=1. The white line represents the interface of the system. The color scale is a log base 10 scale in arbitrary units. Axes are in the units of wavelengths.
Fig. 3. b. Azimuthally polarized light focused onto an interface with NA=.85 and entering medium n=1, and exiting medium n=3.55. The white line represents the interface of the system. The color scale is a log base 10 scale in arbitrary units. Axes are in units of wavelengths.

We now consider an immersion lens (NA=1.4) focusing a radial beam to a dielectric/air interface. Figure 4 shows the total intensity of a radial beam as the interface is moved through focus. As we adjust the position of the interface (which is equivalent to defocusing the beam) we see that the longitudinal field focal spot can extend beyond the interface over a considerable range of defocus positions, but is shifted from the origin. The deviation of the focal spot can be thought of, in a geometrical optics fashion, as a result of spherical aberration due to refraction at the interface. Using the full width at half maximum (FWHM) of the focal spot as a gauge, we may now compare the resolution of various polarizations when focused beyond the dielectric-air interface. Figure 5 shows the FWHM of the intensity of a focused, linearly polarized beam, a focused radial beam (total intensity), and the longitudinal component of a focused radial beam for different values of the interface position.

Fig. 4. (2.1 MB) Movie of the defocusing of a radial beam. The beam is entering through a medium of oil (n=1.518) and exiting into a medium of air (n=1). The white line represents the position of the interface. The color scale is a logarithmic base 10 scale. Axes are in units of wavelengths.[Media 2]
Fig. 5. Full width of half maximum of the focal spot for light of linear and radial polarization (total electric field intensity) and the longitudinal component of the radially polarized beams focused through an immersion lens (NA=1.4). Blue stars correspond to the radially polarized beam (total intensity), red triangles to linearly polarized beams, and green squares represent the longitudinal component of the radial polarization. Both interface positions and the FWHM are given in wavelengths. An interface at a negative position is inside the geometrical focus.

For radially polarized light, the FWHM of the focal spot is somewhat smaller than that of the linear field for a wide range of interface positions before the geometrical focus. However, if we consider only the longitudinal component of the radially polarized beam, it has the smallest FWHM of the other two beam polarizations, over a large range of interface positions before the geometrical focus. We propose, therefore, that linear or nonlinear mechanisms which detect the scattering of longitudinal fields can provide, in air, a resolution equivalent to what is possible with immersion objectives. A mechanism that is strongly responsive to just a longitudinally polarized field is surface harmonic generation (HG). In principle, a surface HG field could be produced at an interface to create an image of the interface, with better resolution than conventional linear or circularly polarized beams. This is likely to be of great importance, for example, in semiconductor mask and wafer inspection.

4. Conclusion

The three dimensional fields of cylindrical vector beams focused through a dielectric interface have been calculated and compared to circularly polarized light. Examples have been given both for air objectives focused on to a high index (e.g. semiconductor) substrate and a dielectric/air immersion system. For the case when the beam is focused through an immersion lens (NA=1.4), the radial beam is of considerable interest because of the rather tight confinement of the longitudinal field and its extent into the air. Such beams are likely to play important roles in a variety of surface imaging applications.

Acknowledgments

We would like to thank Kathleen Youngworth, Professor Lukas Novotny, and Professor Emil Wolf for helpful discussions. This work was supported in part by the Semiconductor Research Corporation under contract 776.001.

References and links

1.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959). [CrossRef]

2.

Hao Ling and Shung-Wu Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984). [CrossRef]

3.

P. Török, P. Varga, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995). [CrossRef]

4.

S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997). [CrossRef]

5.

Lars Egil Helseth, “Roles of polarization, phase, and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

6.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetrical operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs Quantum-well semiconductor-laser,” Appl. Phys. Lett. 60, 1921–1923 (1992). [CrossRef]

7.

R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19, 427–429 (1994). [CrossRef] [PubMed]

8.

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996). [CrossRef] [PubMed]

9.

P. L. Greene and D. G. Hall, “Diffraction characteristics of the azimuthal Bessel-Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996). [CrossRef]

10.

P. L. Greene and D. G. Hall, “Properties and diffraction of vector Bessel-Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998). [CrossRef]

11.

P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999), http://epubs.osa.org/opticsexpress/tocv4n10.htm. [CrossRef] [PubMed]

12.

C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24,1543–1545 (1999). [CrossRef]

13.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234–2239 (1990). [CrossRef] [PubMed]

14.

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE3919 (2000). [CrossRef]

15.

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1949 (1996). [CrossRef] [PubMed]

16.

R. Yamaguchi, T. Nose, and S. Sato, “Liquid-crystal polarizers with axially symmetric properties,” Jpn. J. Appl. Phys. 28, 1730–1731 (1989). [CrossRef]

17.

E. G. Churin, J. Hossfeld, and T. Tschudi, “Polarization configurations with singular point formed by computer generated holograms,” Opt. Commun. 99, 13–17 (1993). [CrossRef]

18.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Exp. 7, 77–87 (2000). [CrossRef]

19.

S. Quabis, R. Dorn, M. Eberler, O. G. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

20.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485, (2000). [CrossRef] [PubMed]

21.

L. Novotny, M.R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254, (2001). [CrossRef] [PubMed]

22.

J. Enderlein, “Theoretical study of detection of a dipole emitter through an objective with high numerical aperture,” Opt. Lett. 25, 634–636 (2000). [CrossRef]

23.

T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, “Polarization spectroscopy of single fluorescent molecules,” J. Phys. Chem. B 103, 6839–6850 (1999) [CrossRef]

OCIS Codes
(110.2990) Imaging systems : Image formation theory
(140.3300) Lasers and laser optics : Laser beam shaping
(260.5210) Physical optics : Photoionization

ToC Category:
Research Papers

History
Original Manuscript: August 2, 2001
Published: November 5, 2001

Citation
David Biss and Thomas Brown, "Cylindrical vector beam focusing through a dielectric interface," Opt. Express 9, 490-497 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-10-490


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References

  1. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II.Structure of the image field in an aplanatic system," Proc. Roy. Soc. A 253, 358-379 (1959). [CrossRef]
  2. Hao Ling and Shung-Wu Lee, "Focusing of electromagnetic waves through a dielectric interface," J. Opt. Soc. Am. A 1, 965-973 (1984). [CrossRef]
  3. P. T�r�k, P.Varga, G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices:structure of the electromagnetic field.I," J. Opt. Soc. Am. A 12, 2136-2144 (1995). [CrossRef]
  4. S. H. Wiersma, P.T�r�k, T. D. Visser, P. Varga, "Comparison of different heories for focusing through a plane interface," J. Opt. Soc. Am. A 14, 1482-1490 (1997). [CrossRef]
  5. Lars Egil Helseth, "Roles of polarization,phase,and amplitude in solid immersion lens systems," Opt. Commun. 191, 161-172 (2001). [CrossRef]
  6. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetrical operation of a concentric-circle-grating,surface-emitting, AlGaAs/GaAs Quantum-well semiconductor-laser," Appl. Phys. Lett. 60, 1921-1923 (1992). [CrossRef]
  7. R. H. Jordan and D. G. Hall, "Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution," Opt. Lett. 19, 427-429 (1994). [CrossRef] [PubMed]
  8. D. G. Hall, "Vector-beam solutions of Maxwell's wave equation," Opt. Lett. 21, 9-11 (1996). [CrossRef] [PubMed]
  9. P. L. Greene and D. G.Hall, "Diffraction characteristics of the azimuthal Bessel-Gauss beam, " J. Opt. Soc. Am. A 13, 962-966 (1996). [CrossRef]
  10. P. L. Greene and D. G. Hall, "Properties and diffraction of vector Bessel-Gauss beams," J. Opt. Soc. Am. A 15, 3020-3027 (1998). [CrossRef]
  11. P. L. Greene and D. G .Hall, "Focal shift in vector beams," Opt. Express 4, 411-419 (1999), http://www.opticsexpress.org/tocv4n10.htm [CrossRef] [PubMed]
  12. C. J. R. Sheppard and S. Saghafi, "Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation," Opt. Lett. 24, 1543-1545 (1999). [CrossRef]
  13. S. C. Tidwell, D. H. Ford, and W. D. Kimura, "Generating radially polarized beams interferometrically," Appl. Opt. 29, 2234-2239 (1990). [CrossRef] [PubMed]
  14. K. S. Youngworth and T. G. Brown, "Inhomogeneous polarization in scanning optical microscopy," Proc. SPIE 3919 (2000). [CrossRef]
  15. M. Stalder and M. Schadt, "Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt. Lett. 21, 1948-1949 (1996). [CrossRef] [PubMed]
  16. R. Yamaguchi, T. Nose, and S. Sato, "Liquid-crystal polarizers with axially symmetric properties," Jpn. J. Appl. Phys. 28, 1730-1731 (1989). [CrossRef]
  17. E. G. Churin, J. Hossfeld, and T. Tschudi, "Polarization configurations with singular point formed by computer generated holograms," Opt.Commun. 99, 13-17 (1993). [CrossRef]
  18. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000), http://www.opticsexpress.org/oearchive/source/22809.htm [CrossRef]
  19. S. Quabis, R. Dorn, M. Eberler, O. G. Gl�ckl, G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000). [CrossRef]
  20. B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485,(2000). [CrossRef] [PubMed]
  21. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001). [CrossRef] [PubMed]
  22. J .Enderlein, "Theoretical study of detection of a dipole emitter through an objective with high numerical aperture," Opt. Lett. 25, 634-636 (2000). [CrossRef]
  23. T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, "Polarization spectroscopy of single fluorescent molecules," J. Phys. Chem. B 103, 6839-6850 (1999) [CrossRef]

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