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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 20 — Sep. 27, 2010
  • pp: 20817–20826
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Beam width of highly-focused radially-polarized fields

R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas  »View Author Affiliations


Optics Express, Vol. 18, Issue 20, pp. 20817-20826 (2010)
http://dx.doi.org/10.1364/OE.18.020817


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Abstract

On the basis of a formal analogy with the irradiance moments, analytical definitions are proposed for the width of both the transverse and the longitudinal component of rotationally-symmetric radially-polarized fields at the focal plane of a high-focusing optical system. The beam width of the whole field is also introduced. The transverse beam size is thus associated with the overall spatial structure of the field. The beam-width definitions are applied to an illustrative example, which enables us to show that, at the focal plane, the power contained within a circle whose radius is given by the proposed beam widths represents the main part of the total power.

© 2010 OSA

1. Introduction

In nanophotonics and nanooptics, a large number of optical devices make use of highly-focused light beams (see, for example, Ref. 1 and references therein). Applications range from optical data storage to confocal microscopy and particle trapping, to mention only some examples. In these cases, the spatial distribution of the polarization of the incident field has been shown to play an essential role. More specifically, radially polarized beams can be particularly useful because, at their focal region, they exhibit spot sizes smaller than the widths shown by conventional linearly polarized fields (see, for instance, Refs. 2

2. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

4

4. D. W. Diehl, R. W. Schoonover, and T. D. Visser, “The structure of focused, radially polarized fields,” Opt. Express 14(7), 3030–3038 (2006). [CrossRef] [PubMed]

). In the experiments, the spot size was introduced as the area that is encircled by the contour line at half the maximum value of the irradiance profile. Consequently, in such framework, the beam width is associated with a local value of the irradiance.

2. Width of the transverse component of the field at the focal plane

Let us consider a monochromatic radially polarized field at the input plane of an aplanatic focusing optical system. As is well known [1

1. L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, Cambridge, 2007).

], the electric field vector E at the focal plane of the system is given by the following expression:
E(r,ϕ)=A0θ002πEi(θ)cosθ(cosθcosφcosθsinφsinθ)exp[ikrsinθcos(ϕφ)]sinθdθdφ,
(4)
where A is a constant, E i is the field amplitude (incident on the system) assumed to be rotationally symmetric around the propagation axis z; r and ϕ denote here the polar coordinates at the focal plane, and the angles θ and θ 0 are represented in Fig. 1
Fig. 1 Illustrating the notation and the geometry of the problem. The value θ0 corresponds to the semi-aperture angle of the aplanatic system.
. In Eq. (4) the vector inside the integral provides the vectorial structure of a pure incident radially polarized field, and the value z = 0 corresponds to the focal plane where we are evaluating E(r,ϕ) .For simplicity, let us write
ρsinθ.
(5)
Accordingly, the transverse and longitudinal field-components at the focal plane become
(ExEy)=A0a02πEi(ρ)(1ρ2)1/4(cosφsinφ)exp[ikrρcos(ϕφ)]ρdρdφ, (6.a)
Ez=A0a02πρEi(ρ)(1ρ2)1/4exp[ikrρcos(ϕφ)]ρdρdφ, (6.b)
where asinθ0 takes into account the angular aperture of the focusing system. For convenience, in the rest of the paper the incident field amplitude E i will be written in the form
Ei(ρ)=ρh(ρ),
(7)
where h(ρ) denotes an arbitrary rotationally-symmetric function.

In the present section we will investigate the transverse part of the field at the focal plane. Note first that Eq. (6.a) can be written in the form
ET(Ex(r,ϕ)Ey(r,ϕ))=2πiA(cosϕsinϕ)f(r),
(8)
with
f(r)=0aEi(ρ)(1ρ2)14J1(krρ)ρdρ,
(9)
where we have used the relation
02π(cosφsinφ)exp[ikrρcos(ϕφ)]dφ=2πi(cosϕsinϕ)J1(krρ).
(10)
Thus, the squared modulus of the transverse component (proportional to the irradiance), at the focal region, reads |ET|2=|Ex|2+|Ey|2=4π2|A|2|f(r)|2, so that

IT(r)02π|ET(r,ϕ)|2dϕ=(2π)3|A|2|f(r)|2.
(11)

On the basis of the formal analogy with the paraxial case (cf. Eq. (1)), we see that the width of the transverse component (symbolized by w T) is closely connected with the width of function f(r). If we try to apply the second-order moment definition to the width of function f, we would find that, in general, the integral diverges. To overcome this trouble and introduce a general width definition for this function, we write f(r) in the form
f(r)=f1(r)+f2(r),
(12)
where
f1(r)=a2krF(a)J2(kra), (13.a)
f2(r)=1kr0aρ2J2(krρ)F(ρ)ρdρ, (13.b)
with

F(ρ)=h(ρ)(1ρ2)14.
(14)

It should be noted that, in a preliminary work, recently reported [14

14. R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, “On the definition of beam width of highly-focused radially-polarized light fields,” Proc. SPIE 7712, 77122K (2010), doi:. [CrossRef]

], we wrote function f(r) in an alternative way. Here we have chosen the analytical expressions given by Eqs. (13) because they will be easier to use for studying the beam-width changes of the focused field at transverse planes around the focal region.

In a similar way to Eq. (1), wT2 can formally be defined as follows:
wT2wT12I1+wT22I2IT,
(15)
where wT12 and wT22 symbolize here the (squared) widths associated to f 1 and f 2,
IT002π|ET(r,ϕ)|2rdrdϕ=0IT(r)rdr=8π3k2|A|20a|ρh(ρ)(1ρ2)14|2ρdρ,
(16)
and

Ij=(2π)3|A|20|fj(r)|2rdr,  j = 1,2.
(17)

Note that, to obtain Eq. (16), use of Parseval theorem has been made. From Eq. (15), we see that the problem of establishing an extended definition of width for this kind of highly-focused vectorial fields reduces to defining, in a suitable way, the quantities wT12 and wT22 .

Concerning wT22 , this width can easily be defined as a second-order moment, namely,

wT22=(2π)3|A|20r2|f2(r)|2rdrI2=(2π)3|A|2k4I20aρ2|F(ρ)ρ|2ρdρ.
(18)

However, if one tries to apply the above moment definition to wT12 , it can be shown that the integral diverges. An alternative definition should then be provided. To do this note first that

|f1(r)|2=a6|F(a)|2[J2(kra)kra]2.
(19)

As occurs, for instance, in diffraction theory, the transverse size of this function could be characterized by the position of the second zero of the Bessel function J 2. We denote this value c 2. Thus, we have
I1wT12=(2π)3|A|24k4a2c22|F(a)|2,
(20)
where we have used the equality
0|f1(r)|2rdr=a6|F(a)|20[J2(kra)kra]2rdr=a44k2|F(a)|2.
(21)
In summary, the (squared) width wT2 of the transverse field takes the analytical form
wT2=1k2I0T[a2c224|F(a)|2+0aρ2|F(ρ)ρ|2ρdρ],
(22)
where
I0T=0a|ρh(ρ)(1ρ2)14|2ρdρ.
(23)
Two special cases should be remarked, which correspond to vanishing either f 1 or f 2 in Eq. (12):
  • a) For those highly-focused fields whose function F(ρ) equals zero at ρ=a , wT2 reduces to the well-known paraxial expression.
  • b) When h(ρ)(1ρ2)14 = constant = h 0, then Fρ=0 , so that f2(r)=0 . Thus we obtain the simple expression

    wT2=h02k2a6c2216|F(a)|2.
    (24)

This case corresponds to an incident beam whose amplitude E i reads

Ei(ρ)=ρh(ρ)=h0ρ(1ρ2)14.
(25)

3. Width of the longitudinal component of the field at the focal plane

I0L=0a|ρ2h(ρ)(1ρ2)14|2ρdρ.
(29)

4. Global width of the whole beam

Let us now consider the whole beam, whose irradiance is proportional to the global (squared) amplitude, i.e., |E|2=|Ex|2+|Ey|2+|Ez|2 . On the basis of the analogy with the second-order irradiance moments (see Eqs. (1-3)), we can introduce wG2 for the whole field in the form
wG2=ITwT2+ILwL2I,
(30)
where I T is given by Eq. (16), and
IL002π|Ez(r,ϕ)|2rdrdϕ,
(31)
with IG=IT+IL . From the beam-width definitions introduced in the above sections for the field components, we finally get
wG2=1k2I0G{a2c224|F(a)|2+c122|G(a)|2+0a[ρ2|F(ρ)ρ|2+|G(ρ)ρ|2]ρdρ},
(32)
where I0G=I0T+I0L , with I0T and I0L defined by Eqs. (23) and (29).

5. Power-content ratio

The validity of the proposed definitions for the beam width should be tested by evaluating the radius of the region (around the z-axis) where the power is concentrated. High enough values of the ratio
Pwj=0wjIj(r)rdr0Ij(r)rdr, j=T,L,G,
(33)
would confirm the appropriateness of the analytical definitions. But before applying them to an illustrative example, an important remark should be pointed out by making use of the Tchebycheff’s inequality [15

15. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

].

Let us consider the simple paraxial (scalar) case, and denote by I(r) and <r2> the (rotationally symmetric) irradiance distribution of the beam profile and its associated second-order moment, respectively. It follows at once
0r2I(r)rdrR0r2I(r)rdrR02R0I(r)rdr,
(34)
where R 0 is an arbitrary positive number. We then have
0R0I(r)rdr0I(r)rdr=1R0I(r)rdr0I(r)rdr11R020r2I(r)rdr0I(r)rdr=1<r2>R02.
(35)
Now, in order to assure that, at least, 75% of the total power is contained within a circular region around the z-axis, we set
0R0I(r)rdr0I(r)rdr1<r2>R02>34,
(36)
and, therefore,
R02<r2>.
(37)
In other words, the radius R 0 of the region that assures 75% of the total power should be twice the root of the second-order irradiance moment.

P2wj=02wjIj(r)rdr0Ij(r)rdr,  j = T, L, G.
(38)

6. Application to an example

Let us now make use of the above analytical definitions, and consider the particular but illustrative case of incident depleted-center field amplitude E i given by
Ei(ρ)=ρh(ρ)=ρexp(f2ρ2ω02)=ρexp(ρ2f02a2),
(39)
where f 0 represents the so-called filling factor, which, for this field, can be defined in the usual way [1

1. L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, Cambridge, 2007).

], namely, f0=ω0(fsinθ0)1 , f being the focal length of the focusing system. Figure 2
Fig. 2 Irradiance (arbitrary units) at the focal plane versus the radial distance r to the propagation axis z (in the figures, the curves have been represented in terms of the dimensionless parameter kr). The figures refer to the transverse component (Fig. 2.a), the longitudinal component (Fig. 2.b), and the whole field (Fig. 2.c). In all the figures, the same scale has been used for ordinates, and the filling factor f 0 has been chosen equal to 1.
plots the irradiance distributions (with f 0 = 1) of the field components and of the whole beam, calculated at the focal plane of the high-focusing system in terms of the radial distance to the z-axis (the field is rotationally symmetric). It is clear that the irradiance is significant for radial distances smaller than λ.

Figure 3
Fig. 3 Dimensionless parameter 2kw at the focal plane, associated with the transverse component (dashed line, 2kwT ), the longitudinal component (dotted line, 2kwL ), and the whole field (continuous line, 2kwG ) versus the filling factor f 0.
shows (a proportionality factor k apart) the widths 2wT , 2wL and 2wG in terms of the filling factor. According with the discussion of Section 5, we have computed the value 2w to assure a significant enough power-content ratio inside a circle whose radius is 2w (see also Fig. 4
Fig. 4 Power-content ratio P 2w (defined by Eq. (40)) in terms of f 0. Dashed line, dotted line and continuous line correspond, respectively, to the transverse component, longitudinal component and whole field.
). It should also be noted that the three curves take the same value for a certain f 0. It should also be noted that, although the initial beam exhibits a depleted-center behavior, the spatial profile of the global beam at the focal region shows a bell-shaped structure.

Finally, Fig. 4 enables to test whether the proposed definitions for the width are physically consistent with the size of the region where the power is concentrated. More specifically, this figure provides the ratio

P2wj=02wjIj(r)rdr0Ij(r)rdr,  j = T, L, G.
(40)

Note that abscises use the same scale in Figs. 3 and 4. As it should be expected, we see from the figure that P 2w is always higher than 88% of the total power. In other words, light energy is mainly focused on a circular region around the z-axis whose radius is twice the respective beam width. It is also interesting to note the small range of variation (less than 10%) of the power-content ratio P 2 w for large variations of the filling factor ( f0[0.5,1.5] ).

7. Conclusions

Associated with the overall spatial structure of a vectorial field, the beam-width definitions based on the irradiance moments have been extended from the paraxial case to highly-focused vectorial fields. More specifically, for incident radially-polarized beams impinging on an aplanatic focusing system, analytical definitions have been proposed for characterizing the beam width of both the transverse and the longitudinal components of the field at the region (focal plane) where the light power is highly concentrated. From the combination of the above definitions, the beam width of the whole field can also be introduced.

Acknowledgements

This work has been supported by the Ministerio de Ciencia e Innovación of Spain, projects FIS2007-63396 and FIS2010-17543. One of the authors (A. M.) acknowledges a FPU grant from the Spanish Ministerio de Educación.

References and links

1.

L. Novotny, and B. Hecht, Principles of Nano-Optics (Cambridge University Press, Cambridge, 2007).

2.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

3.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

4.

D. W. Diehl, R. W. Schoonover, and T. D. Visser, “The structure of focused, radially polarized fields,” Opt. Express 14(7), 3030–3038 (2006). [CrossRef] [PubMed]

5.

S. Lavi, R. Prochaska, and E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27(17), 3696–3703 (1988). [CrossRef] [PubMed]

6.

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttg.) 82, 173–181 (1989).

7.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990). [CrossRef]

8.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), 1027–1049 (1992). [CrossRef]

9.

R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992). [CrossRef]

10.

R. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993). [CrossRef]

11.

P. M. Mejías and R. Martínez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20(7), 660–662 (1995). [CrossRef] [PubMed]

12.

J. R., “Martín de Los Santos, R. Martínez-Herrero and P. M. Mejías, “Parametric characterization of Gaussian beams propagating through active media,” Opt. Quantum Electron. 28, 1021–1027 (1996).

13.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, Berlin, 2009).

14.

R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, “On the definition of beam width of highly-focused radially-polarized light fields,” Proc. SPIE 7712, 77122K (2010), doi:. [CrossRef]

15.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

OCIS Codes
(260.5430) Physical optics : Polarization
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Physical Optics

History
Original Manuscript: July 23, 2010
Revised Manuscript: August 17, 2010
Manuscript Accepted: August 17, 2010
Published: September 16, 2010

Citation
R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, "Beam width of highly-focused radially-polarized fields," Opt. Express 18, 20817-20826 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20817


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References

  1. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, Cambridge, 2007).
  2. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
  3. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
  4. D. W. Diehl, R. W. Schoonover, and T. D. Visser, “The structure of focused, radially polarized fields,” Opt. Express 14(7), 3030–3038 (2006). [CrossRef] [PubMed]
  5. S. Lavi, R. Prochaska, and E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27(17), 3696–3703 (1988). [CrossRef] [PubMed]
  6. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttg.) 82, 173–181 (1989).
  7. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990). [CrossRef]
  8. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), 1027–1049 (1992). [CrossRef]
  9. R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992). [CrossRef]
  10. R. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993). [CrossRef]
  11. P. M. Mejías and R. Martínez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20(7), 660–662 (1995). [CrossRef] [PubMed]
  12. J. R., “Martín de Los Santos, R. Martínez-Herrero and P. M. Mejías, “Parametric characterization of Gaussian beams propagating through active media,” Opt. Quantum Electron. 28, 1021–1027 (1996).
  13. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, Berlin, 2009).
  14. R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, “On the definition of beam width of highly-focused radially-polarized light fields,” Proc. SPIE 7712, 77122K (2010), doi:. [CrossRef]
  15. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

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