Abstract

We propose to use radially, azimuthally and circularly polarized Bessel beams as inhomogeneous illuminating system to unambiguously analyze the vectorial optical response of azo-dye polymers. It is shown that the well-known sensitivity of azo-dye molecules to polarization direction gives rise to surface deformations which are proportional to the longitudinal electric-field component. This property opens a large field of applications in the vectorial analysis of light fields, especially for nano-optics/nanophotonics.

© 2006 Optical Society of America

1. Introduction

It has been shown for more than one decade that azo-polymer films give rise to direct surface modulation when illuminated with visible light in their absorption band [1, 2]. The mechanism of this photoinduced material migration involves repeated photoisomerization cycles of the azobenzene group during light exposure. It is known that the polarization of the incident light beam plays a role in this phenomenon [3]. More recently, radially polarized Bessel beams have been used as test-objects to separate the effects of transverse and longitudinal polarizations in near-field optical imaging process [4]. It has been demonstrated that the commonly used bare fiber-tips play the role of polarization filters as they do not collect the longitudinal electric field component. In this paper, we propose to adapt the technique developed in Ref. [4] to demonstrate that azo-polymer films also provide a polarization filtering (PF) capability through topographical shaping. Such a behavior has direct applications in the vectorial analysis of the electromagnetic fields around structures used in nano-optics. Our study focuses onto the so-called PMMA-DR1 polymer which consists of azo-chromophore DR1 (dispersed red one) grafted into a polymethyl methacrylate matrix (PMMA).

2. Principle and theory

The proposed technique is divided into two steps. First, the azo-dye polymer layer is illuminated by means of a high aperture angle (θ) Bessel beam in order to write a specific imprint onto its free surface. Second, the topography of the exposed area is recorded by usual shear-force detection technique [14]. Let us recall that Bessel beams are the non-diffracting solutions of Maxwell equations [5]. A large number of studies have been devoted to take benefit of these propagation-invariant confined fields in various domain of physics [6, 7, 8]. In our case, the Bessel beams generate field distributions whose square modulus can be described by simple analytical expressions inside the polymer layer (derived from Ref. [9]). We have, for the radially polarized Bessel beam:

ET2=Ccos2θtpT2J12(αr),
EL2=Csin2θtPL2J02(αr),

for the azimuthally polarized Bessel beam:

ET2=Ccos2θtpT2J12(αr),
EL2=0,

for the circularly polarized Bessel beam:

ET2=Cπ[ts+tpTcosθ2J02(αr)+tstpTcosθ2J22(αr)],
EL2=4Cπsin2θtpL2J12(αr),

ET and EL are respectively the transverse and longitudinal field components inside the polymer layer. ET is parallel to the film interfaces whereas EL is perpendicular to them. In the following, |ET |2 and |EL |2 will be called transverse and longitudinal intensities, respectively. α and C are constants (α = (2π/λ)sinθ) and r gives the radial transverse spatial position (parallel to the interfaces) in polar coordinates. Functions Jm are the m-order Bessel functions of the first kind. The photosensitive layer is modelled through coefficients ts , tpT , tpL . In our model, the polymer layer is limited on one side by a semi-infinite glass substrate and on the other side by a semi-infinite air medium. The Bessel beam is incident into the glass substrate and it is transmitted in air through the layer.

 

Fig. 1. Characterization of functions J02 and J12. (a): plots of J02 (solid curve) and J12 (dashed curve); (b) and (c): spatial distributions of J02(r) and J12(r), respectively, calculated in polar coordinates.

Download Full Size | PPT Slide | PDF

In this study, coefficient of J22 in Eq. (5) is sufficiently small so that the contribution of J22 can be neglected in the expression of |ET|2. Therefore, the radially and circularly polarized Bessel beams lead to 3D fields whose transverse intensities are described by functions J12 and J02, respectively, and longitudinal intensities are described by functions J02 and J12, respectively. From Figs. 1(b) and 1(c), it turns out that J02 and J12 are easily identifiable: J02 exhibits a bright central spot whereas J12 shows a central dark spot. Moreover, J02 has maxima around minima of J12, and conversely, J12 has maxima around minima of J02 (Fig. 1(a)).

Such a complementarity makes easy the study of the vectorial optical response of azo-dye polymers. First, these two polarizations fulfill the condition for the generation of deep surface relief patterns. This is due to the fact that circularly and radially polarized Bessel beams can be expanded in circularly and p polarized waves which are known to induce pronounced surface deformations [1, 2, 3]. Second, the discrimination between |ET |2 and |EL |2 in the surface imprint becomes possible and it will be obvious whether the azo-dye polymer exhibits a PF phenomenon. In that case, the study of the polymer response with two complementary excitations is determinant to undoubtedly validate the fact that the surface deformation process records faithfully |EL |2 or |ET |2. Another comparison with the pure 2D-transverse field distribution given by the azimuthally polarized Bessel beam will confirm our conclusions.

Because the argument of the Bessel functions involved in Eqs. (1)-(6) is z non-dependent (α is constant), the light distributions do not spread inside the film. Figure 2 reports the longitudinal cross section of the intensities I = |ET |2 + |EL |2 simulated from Eqs. (1),(2) in radial polarization (Fig. 2(a)), from Eqs. (5),(6) in circular polarization (Fig. 2(b)) and from Eqs. (3),(4) in azimuthal polarization (Fig. 2(c)). Here, the film thickness is equal to 100 nm, θ = 26.5°, λ=514 nm and the refractive index of PMMA-DR1 is assumed to be equal to 1.6 [10]. The non-diffracting behavior of the beam inside the film is here clearly shown. Such a property makes the proposed illumination technique much less sensitive to sample thickness than conventional focused beam techniques for which the spot size in the focal region depends strongly on the longitudinal z-coordinate. Thus, the study of PF in relatively thick samples as well as the study of the relationship between polymer thickness and PF is possible with Bessel beams. Although being non diverging, the intensity distributions do not keep unchanged over the film thickness (Fig. 2). This is explained by the fact that coefficients ts , tpT , tpL are functions of the longitudinal z-coordinate (the polymer layer acts as a longitudinal optical cavity). Therefore, the relative weights of J02 and J12 in the intensity expressions are z dependent. This gives rise to possible contrast variations over the film thickness (Fig. 2(a)).

 

Fig. 2. Longitudinal cross sections of the intensity distributions numerically simulated in the PMMA-DR1 layer with (a) radially polarized, (b) circularly polarized and (c) az-imuthally polarized Bessel beams.

Download Full Size | PPT Slide | PDF

3. Experiments

The experimental setup is depicted in Fig. 3. The experimental parameters are the same as the ones used in Fig. 2. A 100-nanometer-thick PMMA-DR1 layer is deposited by spin-coating onto a microscope cover-glass. The excitation source is an argon laser light beam (λ= 514 nm) which fits the peak absorption wavelength of PMMA-DR1 (around 488 nm [11]).

 

Fig. 3. Recording setup.

Download Full Size | PPT Slide | PDF

The Bessel beams are created by projecting a collimated laser beam onto an axicon-based converging system [12] (Fig. 3(a)). The incident radially and azimuthally polarized beams have have been generated through the optical fiber system described in Ref. [13]. The so-generated Bessel beams have an aperture angle in air slightly smaller than 45°. In this case, they provide longitudinally-invariant submicron light confinements over a distance larger than 1 mm. The exceptionally long depth of field of the axicon is an advantage both to study thick samples and to perform an easy positioning of the photosensitive layer into the beam. There is no necessity to set the film in a given plane as it is the case with focusing objectives. The transverse intensity of the Bessel beam is experimentally mapped by scanning the beam with a common bare fiber local probe, as shown in Fig. 3(a) [4]. The probe is realized by usual heating-pulling technique of a monomode optical fiber.

The exposure setup is reported in Fig. 3(b). The average power-density of the Bessel beams has been set to 32 mW.cm-2. In that case, the maximum intensity inside the polymer film (Fig. 2(a)) has been calculated to a value slightly higher than 40 W.cm-2 with radial polarization (calculations not detailed here). We can expect that the two other polarizations give maximum intensities close to this value. The illumination durations have been limited to 0.5 seconds giving rise to a maximum photon dose of 20 J.cm-2.

The irradiated areas have been subsequently mapped by means of a home-made near-field microscope combined with a shear force detection technique [14] (Fig. 3(c)).

4. Results and discussion

For each radially, circularly and azimuthally polarized Bessel beam, the experimental transverse intensities are reported in Figs. 4(a), 4(e), 4(i), respectively, whereas the longitudinal intensities are simulated in the radial and circular cases (Figs. 4(b), 4(f), respectively). Topography acquisitions are reported in Figs. 4(c), 4(g), 4(j).

We see from Figs. 4(b), 4(c) and 4(f), 4(g) an obvious relationship between the surface deformations and the longitudinal intensity of the illuminating Bessel beams. In radial polarization, the simulated longitudinal intensity (Fig. 4(b)) and the experimentally measured topography (Fig. 4(c)) exhibit a central maximum whereas the experimental transverse intensity provides a central dark spot (Fig. 4(a)). Moreover, the central bump displayed in Fig. 4(c) is of the same size as the central spot of Fig. 4(b). Figure 4(d) confirms the clear similarity between the simulation of |ET |2 (top dashed profile) and the experimental optical inspection (top solid profile) and between the simulation of |EL |2 (bottom dashed profile) and the photoinduced surface deformation (bottom solid profile). In circular polarization, the simulation of |EL |2 (Fig. 4(f)) and the experimental topography acquisition (Fig. 4(g)) exhibit a central minimum whereas |ET |2 presents a central bright spot (Fig. 4(e)). The central relief does not reflect exactly the highly symmetrical annular shape of |EL |2 surrounding the central dark spot (Fig. 4(f)). This can be explained by a slight non-linearity of the polymer layer with respect to the illumination parameters. Nevertheless, as for radial polarization, Fig. 4(h) displays a quasi overlap between the profiles of the experimental optical image (top solid curve) and the simulation of |ET |2 (top dashed curve) as well as between the profiles of |EL |2 simulation (bottom dashed curve) and topography acquisition (bottom solid curve, profile along the direction perpendicular to the central node line).

Figures 4(i)–4(k) depict the case of the azimuthally polarized Bessel beam exhibiting a null |EL |2. Figures 4(j) and 4(k) show that the visibility of the imprint written by the lateral fringes of the Bessel beam does not exceed the noise level of our topography sensing device. It means that the maximum amplitude of the surface deformation is limited to a few nanometers. It is much lower than the one obtained with the two other polarizations since the topography acquisitions exhibit in these cases much higher signal-to-noise ratios (see Figs. 4(c), 4(d) and 4(g), 4(h)).

From these results, we can assert first that the concentrations of matter are mainly located in the maxima of |EL |2 and second that the surface modulation is proportional to the longitudinal intensity (a PF phenomenon is observed). In the following, that will be our first and second conclusions.

 

Fig. 4. Characterization of the optical response of the PMMA-DR1 layer illuminated with a radially polarized Bessel beam ((a)-(d)), a circularly polarized Bessel beam ((e)-(h)) and an azimuthally polarized Bessel beam ((i)-(k)). (a),(e),(i): |ET |2, experiments; (b),(f): |EL |2, theory; (c),(g),(j): topography of the photoinduced surface relief pattern; (d),(h),(k): plots of |ET |2 profiles (theory: top dashed curves, experiments: top solid curves), |EL |2 profiles (bottom dashed curves) and topography acquisition profiles (bottom solid curves). Profile width : 2 μm. (l): comparison between the profiles of the surface imprint and the one calculated with the gradient force model [15], illumination with a radially polarized Bessel beam.

Download Full Size | PPT Slide | PDF

 

Fig. 5. (a): Longitudinal cross section of the intensity simulated in the PMMA-DR1 layer illuminated with a highly convergent radially polarized Bessel beams; (b): Transverse cross section along the white line of (a); (c): Topography of the surface relief pattern. λ = 488 nm.

Download Full Size | PPT Slide | PDF

Our interpretation does not explain the photochemical mechanisms which originate the surface deformation. However, several models have proposed some ways to describe the process [15, 16]. These models use Fresnel diffraction theory to calculate the optical fields. This implies that EL is not taken into account in the interpretation of the photochemical process. Such an approximation is disputable in the description of a polarization sensitive process even whether the study is made in paraxial regime. In our case, it can explain the difference between our experimental surface imprint and the surface relief predicted in Ref. [15] (Fig. 4(l)). Although the maxima of the two curves coincide, which confirms our first conclusion, the modulations do not overlap.

More generally, complete vectorial theory of electromagnetism points out that |ET |2 and |EL |2 exhibit two distributions which do not overlap. For example, the interfering beams used to generate surface relief gratings carry two interference patterns. They are described by complementary square sinusoidal functions exhibiting a π-phase difference to each other. Therefore, the fact that the polymer moves toward the maxima of |EL |2 is consistent with experimental observations of the surface relief grating formation [1]. It gives an alternative explanation to the π-phase difference between |ET |2 and the surface deformation. Here, the azo polymer records faithfully |EL |2, which confirms our second conclusion. The linearly polarized focused fields, described in Ref. [17], exhibit |EL |2 distributions in a two side-lobe structure around the central spot carried by |ET |2. The surface relief patterns generated by such beams show a two-bump structure around a central hollow [18]. From this result, the polymer appears to leave |ET |2 in order to concentrate into the longitudinal intensity maxima. The polymer relief roughly reproduces |EL |2 despite the fact that we observe a strong non-linearity in the center.

Let us stress that care must be taken to avoid the shortcuts between |ET |2 and the total intensity, as suggested by Fresnel theory. For example, it is assumed in Ref. [16] that the material is concentrated in the minima of the light intensity. This is not confirmed in Fig. 5 which displays the surface deformation obtained with a radially polarized highly-convergent Bessel beam (the numerical aperture of the Bessel beam generator is equal to one, λ = 488 nm, the index of refraction of the azo-polymer is equal to 1.55 [10] and the maximum photon dose inside the polymer layer is calculated from experimental parameters to 550 J.cm-2). Topography acquisition has been performed with the usual shear-force detection technique [14]. In that case, the maximum of the light intensity over the film thickness (which corresponds to the maximum of |EL |2) gives rise to the maximum of the surface relief pattern. Figures. 4(j) and 4(k) also show that matter does not move in the minima of the total light intensity.

Our approach has two consequences in the study of azo-dye polymers. First, the models describing the microscopic photochemical reaction must take EL into account. The sensitivity of azobenzene molecules to EL can explain the surface deformation process. Second, our conclusions can be enlarged to any illuminating objects whatever their nature (diffractive objects, fluorescent elements, etc).

5. Conclusion

In this paper, we have shown that azo-dye polymers (PMMA-DR1) have remarkable optical properties as their surface deformation reproduces the longitudinal intensity of the illuminating light field. Such a property makes azo polymer layers opening a wide field of applications in which they can play the role of longitudinal electric-field probe. This new kind of polarization sensitive sensors will offer new possibilities in the vectorial characterization of electromagnetic fields, in particular for nano-optics. Its dielectric nature limiting the risks of probe-to-sample coupling, it can probe either dielectric or metallic nano-objects [19, 20], providing the complementary information missing with usual dielectric bare tips (transverse intensity sensors).

References and links

1. P. Rochon, E. Batalla, and A. Natansohn. “Optically induced surface gratings on azoaromatic polymer films.” Appl. Phys. Lett. 66, 136–138, 1995. [CrossRef]  

2. D.Y. Kim, S.K. Tripathy, L. Li, and J. Kumar. “Laser-induced holographic surface relief gratings on nonlinear optical polymer films.” Appl. Phys. Lett. 66, 1166–1168, 1995. [CrossRef]  

3. X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. “Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films.” Appl. Phys. Lett. 68, 2618–2620, 1996. [CrossRef]  

4. T. Grosjean and D. Courjon. “Polarization filtering induced by imaging systems: Effect on image structure.” Phys. Rev. E 67, 46611, 2003. [CrossRef]  

5. J. Durnin. “Exact solutions for nondiffracting beams. i. the scalar theory.” J. Opt. Soc. Am. A 4:651–654, 1987. [CrossRef]  

6. B. Hafizi, E. Esarey, and P. Sprangle. “Laser-driven acceleration with Bessel beams.” Phys. Rev. E 55, 3539–3545, 1997. [CrossRef]  

7. C. Yu, M.R. Wang, A.J. Varela, and B. Chen. “High-density non-diffracting beam array for optical interconnection.” Optics Commun. 177, 369–376, 2000. [CrossRef]  

8. 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, 1–8, 2001. [CrossRef]  

9. T. Grosjean and D. Courjon. “Immaterial tip concept by light confinement.” J. Microscopy 202, 273–278, 2001. [CrossRef]  

10. Ph. Prêtre, L.-M Wu, A. Knoesen, and J.D. Swalen. “Optical properties of nonlinear optical polymers: a method for calculation.” J. Opt. Soc. Am. A 15, 359–368, 1998. [CrossRef]  

11. R. Piron, E. Toussaere, D. Josse, and J. Zyss. “Enhanced photoinduced birefringence in polymer microcavities.” Appl. Phys. Lett. 77, 2461–2463, 2000. [CrossRef]  

12. J.H. McLeod. “Axicons and their uses.” J. Opt. Soc. Am. 50, 166–169, 1960. [CrossRef]  

13. T. Grosjean, D. Courjon, and M. Spajer. “An all-fiber device for generating radially and other polarized light beams.” Optics Commun. 203, 1–5, 2002. [CrossRef]  

14. K. Karraï and R.D. Grober. “Piezo-electric tuning fork tip-sample distance control for near field optical microscopes.” Ultramicroscopy 61, 197–205, 1995. [CrossRef]  

15. J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. “Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers.” Appl. Phys. Lett. 72, 2096–2098, 1998. [CrossRef]  

16. P. Lefin, C. Fiorini, and J-M. Nunzi. “Anisotropy of the photoinduced translation diffusion of azo-dyes.” Opt. Mater. 9, 323–328, 1998. [CrossRef]  

17. B. Richards and E. Wolf. “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanetic system.” In Proceedings of the Royal Society of London, Series A : Mathematical and Physical Sciences pages 358–379, 1959.

18. S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. “Photoinduced surface deformations on azobenzene polymer films.” J. Appl. Phys. 86, 4498–4508, 1999. [CrossRef]  

19. P.J. Schuck, D.P. Fromm, A. Sundaramurthy, G.S. Kino, and W.E. Moerner. “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas.” Phys. Rev. Lett. 94, 017402, 2005. [CrossRef]   [PubMed]  

20. P. Mühlschlegel, H-J. Eisler, O.J.F. Martin, B. Hecht, and D.W. Pohl. “Resonnant optical antenna.” Science 308, 1607–1609, 2005. [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. P. Rochon, E. Batalla, and A. Natansohn. "Optically induced surface gratings on azoaromatic polymer films." Appl. Phys. Lett. 66, 136-138, 1995.
    [CrossRef]
  2. D.Y. Kim, S.K. Tripathy, L. Li, and J. Kumar. "Laser-induced holographic surface relief gratings on nonlinear optical polymer films." Appl. Phys. Lett. 66, 1166-1168, 1995.
    [CrossRef]
  3. X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. "Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films." Appl. Phys. Lett. 68, 2618-2620, 1996.
    [CrossRef]
  4. T. Grosjean and D. Courjon. "Polarization filtering induced by imaging systems: Effect on image structure." Phys. Rev. E 67, 46611, 2003.
    [CrossRef]
  5. J. Durnin. "Exact solutions for nondiffracting beams. i. the scalar theory." J. Opt. Soc. Am. A 4:651-654, 1987.
    [CrossRef]
  6. B. Hafizi, E. Esarey, and P. Sprangle. "Laser-driven acceleration with Bessel beams." Phys. Rev. E 55, 3539-3545, 1997.
    [CrossRef]
  7. C. Yu, M.R. Wang, A.J. Varela, and B. Chen. "High-density non-diffracting beam array for optical interconnection." Opt. Commun. 177, 369-376, 2000.
    [CrossRef]
  8. 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, 1-8, 2001.
    [CrossRef]
  9. T. Grosjean and D. Courjon. "Immaterial tip concept by light confinement." J. Microscopy 202, 273-278, 2001.
    [CrossRef]
  10. Ph. Prêtre, L.-M Wu, A. Knoesen and J.D. Swalen. "Optical properties of nonlinear optical polymers: a method for calculation." J. Opt. Soc. Am. A 15, 359-368, 1998.
    [CrossRef]
  11. R. Piron, E. Toussaere, D. Josse and J. Zyss. "Enhanced photoinduced birefringence in polymer microcavities." Appl. Phys. Lett. 77, 2461-2463, 2000.
    [CrossRef]
  12. J.H. McLeod. "Axicons and their uses." J. Opt. Soc. Am. 50, 166-169, 1960.
    [CrossRef]
  13. T. Grosjean, D. Courjon, and M. Spajer. "An all-fiber device for generating radially and other polarized light beams." Optics Commun. 203, 1-5, 2002.
    [CrossRef]
  14. K. Karraï and R.D. Grober. "Piezo-electric tuning fork tip-sample distance control for near field optical microscopes." Ultramicroscopy 61, 197-205, 1995.
    [CrossRef]
  15. J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. "Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers." Appl. Phys. Lett. 72, 2096-2098, 1998.
    [CrossRef]
  16. P. Lefin, C. Fiorini, and J-M. Nunzi. "Anisotropy of the photoinduced translation diffusion of azo-dyes." Opt. Mater. 9, 323-328, 1998.
    [CrossRef]
  17. B. Richards and E. Wolf. "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanetic system." In Proceedings of the Royal Society of London, Series A : Mathematical and Physical Sciences pages 358-379, 1959.
  18. S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
    [CrossRef]
  19. P.J. Schuck, D.P. Fromm, A. Sundaramurthy, G.S. Kino, and W.E. Moerner. "Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas." Phys. Rev. Lett. 94, 017402, 2005.
    [CrossRef] [PubMed]
  20. P. Mühlschlegel, H-J. Eisler, O.J.F. Martin, B. Hecht, and D.W. Pohl. "Resonnant optical antenna." Science 308, 1607-1609, 2005.
    [CrossRef] [PubMed]

2005 (2)

P.J. Schuck, D.P. Fromm, A. Sundaramurthy, G.S. Kino, and W.E. Moerner. "Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas." Phys. Rev. Lett. 94, 017402, 2005.
[CrossRef] [PubMed]

P. Mühlschlegel, H-J. Eisler, O.J.F. Martin, B. Hecht, and D.W. Pohl. "Resonnant optical antenna." Science 308, 1607-1609, 2005.
[CrossRef] [PubMed]

2003 (1)

T. Grosjean and D. Courjon. "Polarization filtering induced by imaging systems: Effect on image structure." Phys. Rev. E 67, 46611, 2003.
[CrossRef]

2002 (1)

T. Grosjean, D. Courjon, and M. Spajer. "An all-fiber device for generating radially and other polarized light beams." Optics Commun. 203, 1-5, 2002.
[CrossRef]

2001 (2)

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, 1-8, 2001.
[CrossRef]

T. Grosjean and D. Courjon. "Immaterial tip concept by light confinement." J. Microscopy 202, 273-278, 2001.
[CrossRef]

2000 (2)

C. Yu, M.R. Wang, A.J. Varela, and B. Chen. "High-density non-diffracting beam array for optical interconnection." Opt. Commun. 177, 369-376, 2000.
[CrossRef]

R. Piron, E. Toussaere, D. Josse and J. Zyss. "Enhanced photoinduced birefringence in polymer microcavities." Appl. Phys. Lett. 77, 2461-2463, 2000.
[CrossRef]

1999 (1)

S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
[CrossRef]

1998 (3)

Ph. Prêtre, L.-M Wu, A. Knoesen and J.D. Swalen. "Optical properties of nonlinear optical polymers: a method for calculation." J. Opt. Soc. Am. A 15, 359-368, 1998.
[CrossRef]

J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. "Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers." Appl. Phys. Lett. 72, 2096-2098, 1998.
[CrossRef]

P. Lefin, C. Fiorini, and J-M. Nunzi. "Anisotropy of the photoinduced translation diffusion of azo-dyes." Opt. Mater. 9, 323-328, 1998.
[CrossRef]

1997 (1)

B. Hafizi, E. Esarey, and P. Sprangle. "Laser-driven acceleration with Bessel beams." Phys. Rev. E 55, 3539-3545, 1997.
[CrossRef]

1996 (1)

X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. "Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films." Appl. Phys. Lett. 68, 2618-2620, 1996.
[CrossRef]

1995 (3)

P. Rochon, E. Batalla, and A. Natansohn. "Optically induced surface gratings on azoaromatic polymer films." Appl. Phys. Lett. 66, 136-138, 1995.
[CrossRef]

D.Y. Kim, S.K. Tripathy, L. Li, and J. Kumar. "Laser-induced holographic surface relief gratings on nonlinear optical polymer films." Appl. Phys. Lett. 66, 1166-1168, 1995.
[CrossRef]

K. Karraï and R.D. Grober. "Piezo-electric tuning fork tip-sample distance control for near field optical microscopes." Ultramicroscopy 61, 197-205, 1995.
[CrossRef]

1987 (1)

1960 (1)

Arlt, J.

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, 1-8, 2001.
[CrossRef]

Balasubramanian, S.

S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
[CrossRef]

Batalla, E.

P. Rochon, E. Batalla, and A. Natansohn. "Optically induced surface gratings on azoaromatic polymer films." Appl. Phys. Lett. 66, 136-138, 1995.
[CrossRef]

Bian, S.

S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
[CrossRef]

Chen, B.

C. Yu, M.R. Wang, A.J. Varela, and B. Chen. "High-density non-diffracting beam array for optical interconnection." Opt. Commun. 177, 369-376, 2000.
[CrossRef]

Courjon, D.

T. Grosjean and D. Courjon. "Polarization filtering induced by imaging systems: Effect on image structure." Phys. Rev. E 67, 46611, 2003.
[CrossRef]

T. Grosjean, D. Courjon, and M. Spajer. "An all-fiber device for generating radially and other polarized light beams." Optics Commun. 203, 1-5, 2002.
[CrossRef]

T. Grosjean and D. Courjon. "Immaterial tip concept by light confinement." J. Microscopy 202, 273-278, 2001.
[CrossRef]

Dholakia, K.

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, 1-8, 2001.
[CrossRef]

Durnin, J.

Eisler, H-J.

P. Mühlschlegel, H-J. Eisler, O.J.F. Martin, B. Hecht, and D.W. Pohl. "Resonnant optical antenna." Science 308, 1607-1609, 2005.
[CrossRef] [PubMed]

Esarey, E.

B. Hafizi, E. Esarey, and P. Sprangle. "Laser-driven acceleration with Bessel beams." Phys. Rev. E 55, 3539-3545, 1997.
[CrossRef]

Fiorini, C.

P. Lefin, C. Fiorini, and J-M. Nunzi. "Anisotropy of the photoinduced translation diffusion of azo-dyes." Opt. Mater. 9, 323-328, 1998.
[CrossRef]

Fromm, D.P.

P.J. Schuck, D.P. Fromm, A. Sundaramurthy, G.S. Kino, and W.E. Moerner. "Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas." Phys. Rev. Lett. 94, 017402, 2005.
[CrossRef] [PubMed]

Grober, R.D.

K. Karraï and R.D. Grober. "Piezo-electric tuning fork tip-sample distance control for near field optical microscopes." Ultramicroscopy 61, 197-205, 1995.
[CrossRef]

Grosjean, T.

T. Grosjean and D. Courjon. "Polarization filtering induced by imaging systems: Effect on image structure." Phys. Rev. E 67, 46611, 2003.
[CrossRef]

T. Grosjean, D. Courjon, and M. Spajer. "An all-fiber device for generating radially and other polarized light beams." Optics Commun. 203, 1-5, 2002.
[CrossRef]

T. Grosjean and D. Courjon. "Immaterial tip concept by light confinement." J. Microscopy 202, 273-278, 2001.
[CrossRef]

Hafizi, B.

B. Hafizi, E. Esarey, and P. Sprangle. "Laser-driven acceleration with Bessel beams." Phys. Rev. E 55, 3539-3545, 1997.
[CrossRef]

Hecht, B.

P. Mühlschlegel, H-J. Eisler, O.J.F. Martin, B. Hecht, and D.W. Pohl. "Resonnant optical antenna." Science 308, 1607-1609, 2005.
[CrossRef] [PubMed]

Jiang, X.L.

J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. "Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers." Appl. Phys. Lett. 72, 2096-2098, 1998.
[CrossRef]

X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. "Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films." Appl. Phys. Lett. 68, 2618-2620, 1996.
[CrossRef]

Josse, D.

R. Piron, E. Toussaere, D. Josse and J. Zyss. "Enhanced photoinduced birefringence in polymer microcavities." Appl. Phys. Lett. 77, 2461-2463, 2000.
[CrossRef]

Karraï, K.

K. Karraï and R.D. Grober. "Piezo-electric tuning fork tip-sample distance control for near field optical microscopes." Ultramicroscopy 61, 197-205, 1995.
[CrossRef]

Kim, D.Y.

S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
[CrossRef]

J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. "Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers." Appl. Phys. Lett. 72, 2096-2098, 1998.
[CrossRef]

X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. "Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films." Appl. Phys. Lett. 68, 2618-2620, 1996.
[CrossRef]

D.Y. Kim, S.K. Tripathy, L. Li, and J. Kumar. "Laser-induced holographic surface relief gratings on nonlinear optical polymer films." Appl. Phys. Lett. 66, 1166-1168, 1995.
[CrossRef]

Kino, G.S.

P.J. Schuck, D.P. Fromm, A. Sundaramurthy, G.S. Kino, and W.E. Moerner. "Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas." Phys. Rev. Lett. 94, 017402, 2005.
[CrossRef] [PubMed]

Knoesen, A.

Kumar, J.

S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
[CrossRef]

J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. "Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers." Appl. Phys. Lett. 72, 2096-2098, 1998.
[CrossRef]

X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. "Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films." Appl. Phys. Lett. 68, 2618-2620, 1996.
[CrossRef]

D.Y. Kim, S.K. Tripathy, L. Li, and J. Kumar. "Laser-induced holographic surface relief gratings on nonlinear optical polymer films." Appl. Phys. Lett. 66, 1166-1168, 1995.
[CrossRef]

Lee, T. S.

J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. "Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers." Appl. Phys. Lett. 72, 2096-2098, 1998.
[CrossRef]

Lefin, P.

P. Lefin, C. Fiorini, and J-M. Nunzi. "Anisotropy of the photoinduced translation diffusion of azo-dyes." Opt. Mater. 9, 323-328, 1998.
[CrossRef]

Li, L.

S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
[CrossRef]

J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. "Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers." Appl. Phys. Lett. 72, 2096-2098, 1998.
[CrossRef]

X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. "Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films." Appl. Phys. Lett. 68, 2618-2620, 1996.
[CrossRef]

D.Y. Kim, S.K. Tripathy, L. Li, and J. Kumar. "Laser-induced holographic surface relief gratings on nonlinear optical polymer films." Appl. Phys. Lett. 66, 1166-1168, 1995.
[CrossRef]

Martin, O.J.F.

P. Mühlschlegel, H-J. Eisler, O.J.F. Martin, B. Hecht, and D.W. Pohl. "Resonnant optical antenna." Science 308, 1607-1609, 2005.
[CrossRef] [PubMed]

McLeod, J.H.

Moerner, W.E.

P.J. Schuck, D.P. Fromm, A. Sundaramurthy, G.S. Kino, and W.E. Moerner. "Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas." Phys. Rev. Lett. 94, 017402, 2005.
[CrossRef] [PubMed]

Mühlschlegel, P.

P. Mühlschlegel, H-J. Eisler, O.J.F. Martin, B. Hecht, and D.W. Pohl. "Resonnant optical antenna." Science 308, 1607-1609, 2005.
[CrossRef] [PubMed]

Natansohn, A.

P. Rochon, E. Batalla, and A. Natansohn. "Optically induced surface gratings on azoaromatic polymer films." Appl. Phys. Lett. 66, 136-138, 1995.
[CrossRef]

Nunzi, J-M.

P. Lefin, C. Fiorini, and J-M. Nunzi. "Anisotropy of the photoinduced translation diffusion of azo-dyes." Opt. Mater. 9, 323-328, 1998.
[CrossRef]

Piron, R.

R. Piron, E. Toussaere, D. Josse and J. Zyss. "Enhanced photoinduced birefringence in polymer microcavities." Appl. Phys. Lett. 77, 2461-2463, 2000.
[CrossRef]

Pohl, D.W.

P. Mühlschlegel, H-J. Eisler, O.J.F. Martin, B. Hecht, and D.W. Pohl. "Resonnant optical antenna." Science 308, 1607-1609, 2005.
[CrossRef] [PubMed]

Prêtre, Ph.

Rochon, P.

P. Rochon, E. Batalla, and A. Natansohn. "Optically induced surface gratings on azoaromatic polymer films." Appl. Phys. Lett. 66, 136-138, 1995.
[CrossRef]

Schuck, P.J.

P.J. Schuck, D.P. Fromm, A. Sundaramurthy, G.S. Kino, and W.E. Moerner. "Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas." Phys. Rev. Lett. 94, 017402, 2005.
[CrossRef] [PubMed]

Shivshanvar, V.

X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. "Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films." Appl. Phys. Lett. 68, 2618-2620, 1996.
[CrossRef]

Soneson, J.

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, 1-8, 2001.
[CrossRef]

Spajer, M.

T. Grosjean, D. Courjon, and M. Spajer. "An all-fiber device for generating radially and other polarized light beams." Optics Commun. 203, 1-5, 2002.
[CrossRef]

Sprangle, P.

B. Hafizi, E. Esarey, and P. Sprangle. "Laser-driven acceleration with Bessel beams." Phys. Rev. E 55, 3539-3545, 1997.
[CrossRef]

Sundaramurthy, A.

P.J. Schuck, D.P. Fromm, A. Sundaramurthy, G.S. Kino, and W.E. Moerner. "Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas." Phys. Rev. Lett. 94, 017402, 2005.
[CrossRef] [PubMed]

Swalen, J.D.

Toussaere, E.

R. Piron, E. Toussaere, D. Josse and J. Zyss. "Enhanced photoinduced birefringence in polymer microcavities." Appl. Phys. Lett. 77, 2461-2463, 2000.
[CrossRef]

Tripathy, S.

S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
[CrossRef]

J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. "Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers." Appl. Phys. Lett. 72, 2096-2098, 1998.
[CrossRef]

X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. "Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films." Appl. Phys. Lett. 68, 2618-2620, 1996.
[CrossRef]

Tripathy, S.K.

D.Y. Kim, S.K. Tripathy, L. Li, and J. Kumar. "Laser-induced holographic surface relief gratings on nonlinear optical polymer films." Appl. Phys. Lett. 66, 1166-1168, 1995.
[CrossRef]

Varela, A.J.

C. Yu, M.R. Wang, A.J. Varela, and B. Chen. "High-density non-diffracting beam array for optical interconnection." Opt. Commun. 177, 369-376, 2000.
[CrossRef]

Wang, M.R.

C. Yu, M.R. Wang, A.J. Varela, and B. Chen. "High-density non-diffracting beam array for optical interconnection." Opt. Commun. 177, 369-376, 2000.
[CrossRef]

Williams, J.M.

S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
[CrossRef]

Wright, E.M.

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, 1-8, 2001.
[CrossRef]

Wu, L.-M

Yu, C.

C. Yu, M.R. Wang, A.J. Varela, and B. Chen. "High-density non-diffracting beam array for optical interconnection." Opt. Commun. 177, 369-376, 2000.
[CrossRef]

Zyss, J.

R. Piron, E. Toussaere, D. Josse and J. Zyss. "Enhanced photoinduced birefringence in polymer microcavities." Appl. Phys. Lett. 77, 2461-2463, 2000.
[CrossRef]

Appl. Phys. Lett. (5)

P. Rochon, E. Batalla, and A. Natansohn. "Optically induced surface gratings on azoaromatic polymer films." Appl. Phys. Lett. 66, 136-138, 1995.
[CrossRef]

D.Y. Kim, S.K. Tripathy, L. Li, and J. Kumar. "Laser-induced holographic surface relief gratings on nonlinear optical polymer films." Appl. Phys. Lett. 66, 1166-1168, 1995.
[CrossRef]

X.L. Jiang, L. Li, J. Kumar, D.Y. Kim, V. Shivshanvar, and S. Tripathy. "Polarization dependent recordings of surface relief gratings on azobenzene containing polymer films." Appl. Phys. Lett. 68, 2618-2620, 1996.
[CrossRef]

J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T. S. Lee, and S. Tripathy. "Gradient force: The mechanism for surface relief grating formation in azobenzene functionalized polymers." Appl. Phys. Lett. 72, 2096-2098, 1998.
[CrossRef]

R. Piron, E. Toussaere, D. Josse and J. Zyss. "Enhanced photoinduced birefringence in polymer microcavities." Appl. Phys. Lett. 77, 2461-2463, 2000.
[CrossRef]

J. Appl. Phys. (1)

S. Bian, J.M. Williams, D.Y. Kim, L. Li, S. Balasubramanian, J. Kumar, and S. Tripathy. "Photoinduced surface deformations on azobenzene polymer films." J. Appl. Phys. 86, 4498-4508, 1999.
[CrossRef]

J. Microscopy (1)

T. Grosjean and D. Courjon. "Immaterial tip concept by light confinement." J. Microscopy 202, 273-278, 2001.
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

C. Yu, M.R. Wang, A.J. Varela, and B. Chen. "High-density non-diffracting beam array for optical interconnection." Opt. Commun. 177, 369-376, 2000.
[CrossRef]

Opt. Mater. (1)

P. Lefin, C. Fiorini, and J-M. Nunzi. "Anisotropy of the photoinduced translation diffusion of azo-dyes." Opt. Mater. 9, 323-328, 1998.
[CrossRef]

Optics Commun. (1)

T. Grosjean, D. Courjon, and M. Spajer. "An all-fiber device for generating radially and other polarized light beams." Optics Commun. 203, 1-5, 2002.
[CrossRef]

Phys. Rev. A (1)

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, 1-8, 2001.
[CrossRef]

Phys. Rev. E (2)

T. Grosjean and D. Courjon. "Polarization filtering induced by imaging systems: Effect on image structure." Phys. Rev. E 67, 46611, 2003.
[CrossRef]

B. Hafizi, E. Esarey, and P. Sprangle. "Laser-driven acceleration with Bessel beams." Phys. Rev. E 55, 3539-3545, 1997.
[CrossRef]

Phys. Rev. Lett. (1)

P.J. Schuck, D.P. Fromm, A. Sundaramurthy, G.S. Kino, and W.E. Moerner. "Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas." Phys. Rev. Lett. 94, 017402, 2005.
[CrossRef] [PubMed]

Science (1)

P. Mühlschlegel, H-J. Eisler, O.J.F. Martin, B. Hecht, and D.W. Pohl. "Resonnant optical antenna." Science 308, 1607-1609, 2005.
[CrossRef] [PubMed]

Ultramicroscopy (1)

K. Karraï and R.D. Grober. "Piezo-electric tuning fork tip-sample distance control for near field optical microscopes." Ultramicroscopy 61, 197-205, 1995.
[CrossRef]

Other (1)

B. Richards and E. Wolf. "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanetic system." In Proceedings of the Royal Society of London, Series A : Mathematical and Physical Sciences pages 358-379, 1959.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Characterization of functions J02 and J12. (a): plots of J02 (solid curve) and J12 (dashed curve); (b) and (c): spatial distributions of J02(r) and J12(r), respectively, calculated in polar coordinates.

Fig. 2.
Fig. 2.

Longitudinal cross sections of the intensity distributions numerically simulated in the PMMA-DR1 layer with (a) radially polarized, (b) circularly polarized and (c) az-imuthally polarized Bessel beams.

Fig. 3.
Fig. 3.

Recording setup.

Fig. 4.
Fig. 4.

Characterization of the optical response of the PMMA-DR1 layer illuminated with a radially polarized Bessel beam ((a)-(d)), a circularly polarized Bessel beam ((e)-(h)) and an azimuthally polarized Bessel beam ((i)-(k)). (a),(e),(i): |ET |2, experiments; (b),(f): |EL |2, theory; (c),(g),(j): topography of the photoinduced surface relief pattern; (d),(h),(k): plots of |ET |2 profiles (theory: top dashed curves, experiments: top solid curves), |EL |2 profiles (bottom dashed curves) and topography acquisition profiles (bottom solid curves). Profile width : 2 μm. (l): comparison between the profiles of the surface imprint and the one calculated with the gradient force model [15], illumination with a radially polarized Bessel beam.

Fig. 5.
Fig. 5.

(a): Longitudinal cross section of the intensity simulated in the PMMA-DR1 layer illuminated with a highly convergent radially polarized Bessel beams; (b): Transverse cross section along the white line of (a); (c): Topography of the surface relief pattern. λ = 488 nm.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

E T 2 = C cos 2 θ t pT 2 J 1 2 ( αr ) ,
E L 2 = C sin 2 θ t PL 2 J 0 2 ( αr ) ,
E T 2 = C cos 2 θ t pT 2 J 1 2 ( αr ) ,
E L 2 = 0 ,
E T 2 = C π [ t s + t pT cos θ 2 J 0 2 ( αr ) + t s t pT cos θ 2 J 2 2 ( αr ) ] ,
E L 2 = 4 C π sin 2 θ t pL 2 J 1 2 ( αr ) ,

Metrics