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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 18 — Aug. 27, 2012
  • pp: 20228–20237
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Non-paraxial TM01 and TE01 from Laguerre-Gauss angular spectrum

Pierre-André Bélanger and Simon Thibault  »View Author Affiliations


Optics Express, Vol. 20, Issue 18, pp. 20228-20237 (2012)
http://dx.doi.org/10.1364/OE.20.020228


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Abstract

We demonstrate that a Laguerre-Gauss spectrum of plane waves distribution optimize the variance of the spectrum-bandwidth product. In the space domain, the axial Ez (TM01) and the azimuthal Eϕ (TE01) have also a Laguerre-Gauss profile that describes correctly some experimental published and calculated results in the focal plane.

© 2012 OSA

1. Introduction

In several applications, a sharply focused light beam is required [1

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

3

3. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009). [CrossRef] [PubMed]

]. It has been demonstrated that a radially polarized and an azimuthally polarized light beam can generate respectively a very sharp axial electrical field and a sharp axial magnetic field [4

4. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000). [CrossRef]

,5

5. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002). [PubMed]

]. Such axial electrical and magnetic fields find application in classical optics as well as in nonlinear optics [6

6. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef] [PubMed]

,7

7. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

]. These two types of polarization require the generation of laser TM01 and TE01 beam. Such beam can be generated after passing a Gaussian laser beam through a polarization converter and the field components are evaluated at the focus of the optical system using the well-known Richards–Wolf [8

8. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253(1274), 358–379 (1959). [CrossRef]

] three propagation integrals..

In this paper we use first the Maxwell equations to write down each component of the field as a spectrum of plane wave. We optimize the spectrum bandwidth product (σ02σ^02) defined by their variance to obtain an ideal spectrum distribution. Thereafter, we calculate in closed form the main field components with the optimized Laguerre-Gauss spectrum profiles and comparison of our calculated distribution is made with some experimental result of Dorn et al. [1

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

], Dehez et al. [3

3. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009). [CrossRef] [PubMed]

], Jia et al [9

9. B. Jia, X. Gan, and M. Gu, “Direct measurement of a radially polarized focused evanescent feield facilitated by a single LCD,” Opt. Express 13(18), 6821–6827 (2005). [CrossRef]

] and Hao et al. [10

10. B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15(6), 3550–3556 (2007). [CrossRef] [PubMed]

].

2. Theoretical

In cylindrical coordinates, the axial components of the field Ez and Hz obey the wave equation and therefore they can be written as an angular spectrum A(τ) of plane waves. Both distributions are given at the Eq. (1) where k is the wave number in image space.
EzorHz=[0A(τ)eikz1τ2Jo(krτ)τdτ]
(1)
Using the Maxwell equations and η as the impedance in vacuum, the transverse component for a TM0 and TE0 distribution is given respectively by the Eqs. (2) (TM) and (3) (TE).

TM{Er=i0A(τ)1τ2eikz1τ2J1(krτ)dτHφ=iη0A(τ)eikz1τ2J1(krτ)dτ
(2)
TE{Hr=i0A(τ)1τ2eikz1τ2J1(krτ)dτEϕ=iη0A(τ)eikz1τ2J1(krτ)dτ
(3)

We are now interested to find out a spectrum A(τ) that will optimize the spectrum bandwidth product of the electric fields. We will first consider the transverse field Eϕ which will ensure a finite energy for all fields.

The variance of the spectrum is given by Eq. (4) and the variance of the beam is given by Eq. (5) with the help of the Dirac delta function for Bessel function (Eq. (6)) [11

11. I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, Sixth Edition (Academic Press, 2000) Chap. 6.5.

]. For simplicity, A(τ) is noted only as ‘A’ in the equations.
σ^02=k20A2τ2dττ0A2dττ
(4)
σ02=0|Eϕ|r2(rdr)0|Eϕ|2(rdr)
(5)
0Jn(ar)Jn(br)rdr=1aδ(ab)
(6)
Equation (5) can also be expressed in term of the spectrum distribution and the equation is given by Eq. (7).
k2σ02=0(dAdτ)2dττ0A2dττ
(7)
Equation (7) is obtained using krJ1(krτ)=J0(krτ)τ and integrating by part Eq. (3) for Eϕ and with the condition A(τ) = 0 at τ = 0 and τ = ∞. The spectrum distribution is read at z = 0 and supposed to be real. We normalized the spectrum variance such that the space and spectrum bandwidth product for a pure Gaussian beam is unity.

Now our goal is to find the minimum spatial variance for a fixed total energy and a fixed spectrum variance. Using Lagrange multipliers (Λ0andΛ1) we can implement these two constraints and these conditions can be written as a variational problem expressed by the Eq. (8). The first term is the function to minimize (spatial variance), the second term is the energy and the last is the spectrum variance.

[0[(dAdτ)2dττ+Λ00A2dττ+Λ10A2τ2dττ]]=0
(8)

The Euler-Lagrange equation [12

12. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), chap. 17.6.

]. for Eq. (8) leads to the differential Eq. (9) for the spectrum distribution A(τ).
ddτ(1τdAdτ)Λ0AτΛ1Aτ=0
(9)
Solution of the differential Eq. (9) that are finite at the origin and at infinity are given in Eq. (10) where Lm1(x) is the Laguerre polynomial of order (1) and (m). The Lagrange multipliers (9) are also given by the Eq. (10).
Am(τ)=A0τ2eτ22f2Lm(1)(τ2f2)withm=0,1,2,...Λ0f2=4(m+1),Λ1=1f4
(10)
The parameter f 2 introduced here specifies the width of the angular spectrum. For example, when m = 0, the variance of the spectrum is given by the Eq. (11).
σ^02=k20τ2A2τdτ0A2τdτ=2f2k2
(11)
The spectrum distribution forms an orthogonal set of functions with the orthogonally given in Eq. (12).
0eτ2f2Lm(1)(τ2f2)Ln(1)(τ2f2)τ3dτ=0nm=f42(n+1)n=m
(12)
This Laguerre-Gauss spectrum has already been used for calculating the focal spot of radially polarized laser beams by Kozawa and Sato [13

13. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007). [CrossRef] [PubMed]

] while using the Richards-Wolf [8

8. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253(1274), 358–379 (1959). [CrossRef]

] integrals. In the space domain and equivalent Laguerre-Gaussian beams have been derived in the paraxial approximation as self-similar beam polarized propagation [14

14. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998). [CrossRef]

].

For the TM mode, the transverse field Er will have the same extremum spectrum bandwidth product as Eq. (10) while A(τ) is replaced by A(τ)1τ2. However this introduction of the square root term will yield that the energy in the Ez field is infinite. The spectral distribution A(τ) in Eq. (10) for both TE and TM beams will be used in the section 3.

According to Eq. (10), we obtain not a single angular spectrum that minimise the spatial variance but a family of angular spectra. For Eϕ and Ez, the spectrum-bandwidth product is 4(m+1)2 and 3(m+1)2 respectively. Consequently, the spectrum-bandwidth product is minimal for the fundamental mode (m = 0).

3. Comparison to experimental results

In their experimental set up, Dorn et al. [1

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

] have generated a TM01 and TE01 beam distribution after passing a quasi TEM00 Gaussian beam through a four half-wave plates polarization corrector. The output beam is then close to a TM01 or TE01 beam profile depending on the orientation of the polarizers. After strongly focusing the beam, they measured the Ez distribution of the TM01 and the Eϕ distribution of the TE01 components in the space domain. The focusing generates a beam showing a smaller variance σ02 (larger f 2) and we estimated that their observed beam distribution can be close to the optimum beam profile according to Eq. (10). Using the relation Eq. (13) for the Laguerre polynomial [15

15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publication Inc., New York, 1964), 22.12.7.

], we calculate the distribution of the Ez and Eϕ space component for the spectrum of Eq. (10).
xLm(1)(x)=(m+1)[Lm(x)Lm+1(x)]
(13)
After integration the Ez component of the TM mode and the Eϕ component of the TE mode are given respectively by Eqs. (14) and (15) where c0,c1 (normalization constant).
Ez=c0(1)m(m+1)exp[(kfr)22][Lm((kfr)2)+Lm+1((kfr)2)]
(14)
Eϕ=c1(1)mexp[(kfr)22](kfr)Lm(1)((kfr)2)
(15)
Finally, the normalized distribution of the fundamental mode (m = 0) are given in Eqs. (16) and (17) respectively.
Ez(r,0)E0=exp[(kfr)22](1(kfr)22)
(16)
Eϕ(r,0)E0=2(kfr)·exp[12]·exp[(kfr)22]
(17)
For the Eϕ(r,0) we have already obtained that the variance width is σ02|ϕ=2k2f2and for the Ez(r,0) field it is possible to show that the variance width is σ02|z=1k2f2 . For simplicity, the Eqs. (16) and (17) is written using the Ez(r,0) spatial variance σ02 to Eqs. (18) and (19).
Ez(r,0)E0=exp[r22σ02][1r22σ02]
(18)
Eϕ(r,0)E0=2e12(rσ0)exp[r22σ02]
(19)
The normalized Eqs. (18) and (19) are plotted in the Fig. 1(a)
Fig. 1 Normalized Ez from Eq. (18) (a) and normalized Eϕ fields from Eq. (19) (b).
and 1(b) respectively.

We can also calculate the spot size component (Eq. (18)). The spot size is defined as the area (not diameter) limited by the full width half maximum (FWHM) intensity value [1

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

]. The spot size is given by Eq. (20).
SpotSizeFWHM=0.0263λ2f21.038σ02
(20)
This is smaller by a factor of 2 then the spot size of a pure Gaussian beam.

In another experimental setup, Dehez et al. [3

3. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009). [CrossRef] [PubMed]

] have improved the resolution of a two-photon microscope using a TM01 laser beam. The measured spot size was 0.15λ2 for Ez longitudinal field. The profile of the distribution shown in this paper is also compatible with our predicted Ez field for the fundamental TM01 mode. The profile for the TE01 distribution is also close to our predicted field. However, their experimental results do not clearly show the secondary peak of the field and within the experimental error, we cannot conclude that they generated a fundamental TM01 or TE01 beams.

In recent experimental measurement, Hao and Leger [10

10. B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15(6), 3550–3556 (2007). [CrossRef] [PubMed]

] recorded the focal pattern of the radially polarized light in a photoresist material. Their experimental points (see Fig. 5(b) of Ref [10

10. B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15(6), 3550–3556 (2007). [CrossRef] [PubMed]

].) seems to follow the two peaks of the Laguerre-Gauss profile (21).

4. Discussion

In recent publications (see for example ref [1

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

], [7

7. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

], [13

13. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007). [CrossRef] [PubMed]

], and [16

16. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31(6), 820–822 (2006). [CrossRef] [PubMed]

]) the TM0 and TE0 distributions have been calculated using three propagation integrals derived by Richards and Wolf [8

8. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253(1274), 358–379 (1959). [CrossRef]

], and very similar profiles as the ones obtained here has been formed. The Richards and Wolf results have been developed after propagating plane waves (E = ηH) represented by geometrical rays through an aplanatic focusing system. Here we have propagated the electric and magnetic field accordingly to Maxwell equations and we have not specified any particular focusing system. Richards and Wolf [8

8. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253(1274), 358–379 (1959). [CrossRef]

] made their first analysis for an aplanatic focusing system and they used the sine condition for energy. This condition yields the term (cos1/2θ) in their integral while here the corresponding term is (cosθ) (in our notation cos2θ = 1-τ2). Richards and Wolf [8

8. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253(1274), 358–379 (1959). [CrossRef]

] assumed that the magnetic fields to be proportional to the electric one, here as shown on Eqs. (2) and (3) they are not exactly proportional. However for the paraxial limit (τ2<<1) the two developments are identical. Our analysis here was made assuming that we were at the focus of an optical system (real spectrum). Propagation of the spectrum for z>0 can be made numerically or after making analytically integration required by each fields [1

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

3

3. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009). [CrossRef] [PubMed]

]. For the Laguerre-Gauss spectrum (10) introduced here we can use the following Hermite Hn(x) generating function (24) [15

15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publication Inc., New York, 1964), 22.12.7.

].
exp[12f2(1τ2)ikz1τ2]=n=0(12f2)n2n!Hn(kfz2)(1τ2)n2
(24)
We generally neglect the contribution of the evanescent waves after propagation and for cosθ ≤ π/2 (τ2≤1) all the integration can be made with a well-known integral [15

15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publication Inc., New York, 1964), 22.12.7.

] of Eq. (25).

0π/2Jμ(xsint)sinμ+1(t)cos2υ+1dt=2υΓ(1+υ)xυ+1Jμ+υ+1(x)
(25)

Finally each components of the TM or the TE field can be written in closed form as a summation of Bessel function Jn+2(kr)time a Hermite function Hn(kfz2).

5. Conclusion

In this paper we have shown that solving directly the Maxwell equation for a TM or a TE distribution in an angular spectrum of plane waves in cylindrical coordinates all the field components can be written in term of these integrals that are similar to the integral derived by Richards-Wolf [8

8. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253(1274), 358–379 (1959). [CrossRef]

]. And after requiring that the spectrum bandwidth product of the transverse electrical field to be minimum in variance, a Gauss-Laguerre profile for the spectrum is recovered.

In the paraxial approximation it is possible to show that each components of the magnetic and electrical field propagate according to a pure self-similar Gauss-Laguerre beam as first shown by Nesterov and Niziev [17

17. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). [CrossRef]

].

Acknowledgments

This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the NSERC Industrial Research Chair in Lens Design.

References and links

1.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

2.

D. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9(10), 490–497 (2001). [CrossRef] [PubMed]

3.

H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009). [CrossRef] [PubMed]

4.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000). [CrossRef]

5.

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002). [PubMed]

6.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef] [PubMed]

7.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

8.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253(1274), 358–379 (1959). [CrossRef]

9.

B. Jia, X. Gan, and M. Gu, “Direct measurement of a radially polarized focused evanescent feield facilitated by a single LCD,” Opt. Express 13(18), 6821–6827 (2005). [CrossRef]

10.

B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express 15(6), 3550–3556 (2007). [CrossRef] [PubMed]

11.

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, Sixth Edition (Academic Press, 2000) Chap. 6.5.

12.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), chap. 17.6.

13.

Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007). [CrossRef] [PubMed]

14.

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998). [CrossRef]

15.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publication Inc., New York, 1964), 22.12.7.

16.

Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31(6), 820–822 (2006). [CrossRef] [PubMed]

17.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). [CrossRef]

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

ToC Category:
Physical Optics

History
Original Manuscript: August 8, 2012
Revised Manuscript: August 8, 2012
Manuscript Accepted: August 16, 2012
Published: August 20, 2012

Citation
Pierre-André Bélanger and Simon Thibault, "Non-paraxial TM01 and TE01 from Laguerre-Gauss angular spectrum," Opt. Express 20, 20228-20237 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20228


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References

  1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett.91(23), 233901 (2003). [CrossRef] [PubMed]
  2. D. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9(10), 490–497 (2001). [CrossRef] [PubMed]
  3. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett.34(23), 3601–3603 (2009). [CrossRef] [PubMed]
  4. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000). [CrossRef]
  5. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10(7), 324–331 (2002). [PubMed]
  6. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001). [CrossRef] [PubMed]
  7. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000). [CrossRef] [PubMed]
  8. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A253(1274), 358–379 (1959). [CrossRef]
  9. B. Jia, X. Gan, and M. Gu, “Direct measurement of a radially polarized focused evanescent feield facilitated by a single LCD,” Opt. Express13(18), 6821–6827 (2005). [CrossRef]
  10. B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express15(6), 3550–3556 (2007). [CrossRef] [PubMed]
  11. I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, Sixth Edition (Academic Press, 2000) Chap. 6.5.
  12. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), chap. 17.6.
  13. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A24(6), 1793–1798 (2007). [CrossRef] [PubMed]
  14. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A15(10), 2705–2711 (1998). [CrossRef]
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publication Inc., New York, 1964), 22.12.7.
  16. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett.31(6), 820–822 (2006). [CrossRef] [PubMed]
  17. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys.33(15), 1817–1822 (2000). [CrossRef]

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