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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9201–9212
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Focusing a TM01 beam with a slightly tilted parabolic mirror

Alexandre April, Pierrick Bilodeau, and Michel Piché  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9201-9212 (2011)
http://dx.doi.org/10.1364/OE.19.009201


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Abstract

A parabolic mirror illuminated with an incident collimated beam whose axis of propagation does not exactly coincide with the axis of revolution of the mirror shows distortion and strong coma. To understand the behavior of such a focused beam, a detailed description of the electric field in the focal region of a parabolic mirror illuminated with a beam having a nonzero angle of incidence is required. We use the Richards–Wolf vector field equation to investigate the electric energy density distribution of a beam focused with a parabolic mirror. The explicit aberration function of this focused field is provided along with numerically calculated electric energy densities in the focal region for different angles of incidence. The location of the peak intensity, the Strehl ratio and the full-width at half-maximum as a function of the angle of incidence are given and discussed. The results confirm that the focal spot of a strongly focused beam is affected by severe coma, even for very small tilting of the mirror. This analysis provides a clearer understanding of the effect of the angle of incidence on the focusing properties of a parabolic mirror as such a focusing device is of growing interest in microscopy.

© 2011 OSA

1. Introduction

Reaching the smallest possible focal spot size is useful in many applications, such as in high-resolution microscopy [1

1. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM(01) laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009). [CrossRef] [PubMed]

] and lithography [2

2. H. Ichikawa and H. Kikuta, “Numerical feasibility study of the fabrication of subwavelength structure by mask lithography,” J. Opt. Soc. Am. A 18(5), 1093–1100 (2001). [CrossRef]

]. Such sharper focal spots can be generated when the incident optical beam is focused in such a way that the rays are directed toward the focal point from nearly all directions. This requires a focusing element that subtends a solid angle approaching 4π srad, such as a high numerical aperture parabolic mirror [3

3. N. Bokor and N. Davidson, “4π focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008). [CrossRef]

,4

4. A. April and M. Piché, “4π Focusing of TM(01) beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010). [CrossRef] [PubMed]

]. Nonetheless, the parabolic reflector is rarely exploited for laser focusing, probably because of the difficulty in manufacturing high-aperture paraboloidal mirrors of sufficient optical quality. In fact, a paraboloidal surface showing deviations from ideal curvature that are smaller than the laser wavelength is needed to achieve a diffraction-limited focal spot. Still, it has been verified that a confocal microscope using a high numerical aperture parabolic mirror can deliver excellent images with only small deviations from the calculated focal electric energy distribution [5

5. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]

,6

6. A. Drechsler, M. A. Lieb, C. Debus, A. J. Meixner, and G. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9(12), 637–644 (2001). [CrossRef] [PubMed]

]. Therefore, it appears that a parabolic mirror objective has a promising future as the focusing element in high-resolution confocal microscopy.

Focusing a radially polarized beam with a parabolic mirror can further reduce the focal area. In fact, it has been pointed out that smaller spot sizes can be achieved with a radially polarized beam instead of a linearly polarized beam [7

7. 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]

10

10. T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007). [CrossRef]

]. On the one hand, when a linearly polarized beam is strongly focused, the longitudinal component of its electric field is not cylindrically symmetric and, as a consequence, the focal spot becomes asymmetrically deformed and elongated in the direction of the polarization. On the other hand, when a radially polarized beam is tightly focused, the radial symmetry of the electromagnetic fields yields a perfectly symmetric focal spot with a strong longitudinal electric field component centered on the optical axis, resulting in a reduction of the area of the focal region. Furthermore, it has been shown experimentally and theoretically that parabolic mirrors can focus a radially polarized laser beam to a considerably tighter focal spot than aplanatic lenses do [11

11. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004). [CrossRef] [PubMed]

,12

12. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008). [CrossRef] [PubMed]

]. Thus the use of a high numerical aperture parabolic mirror with a radially polarized beam is very well suited to achieve minimum focal spot size [13

13. C. Debus, M. A. Lieb, A. Drechsler, and A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003). [CrossRef] [PubMed]

].

A collimated beam whose axis of propagation is parallel to the axis of revolution of a parabolic mirror is, by definition, perfectly focused at the focal point of the mirror without any aberration in the geometrical optics approximation. However, when the propagation axis of the incident collimated beam is slightly tilted with respect to the mirror axis, the rays are no longer focused at a single point (Fig. 1
Fig. 1 An exact ray tracing shows that, when the incident rays of a collimated beam are not perfectly parallel to the axis of revolution of the mirror, the focused rays form a catacaustic.
): rather, they form a caustic (more precisely a catacaustic), which is defined as the envelope of light rays reflected by the mirror [14

14. E. W. Weisstein, “Catacaustic.” From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/Catacaustic.html.

,15

15. J. E. Howard, “Imaging properties of off-axis parabolic mirrors,” Appl. Opt. 18(15), 2714–2722 (1979). [CrossRef] [PubMed]

]. When the angle of incidence of the illuminating collimated beam with respect to the axis of revolution of the parabolic mirror is very small, the electromagnetic field distribution can be regarded as a focal spot modestly affected by optical aberrations. The amplitude distribution of the fields of the focused beam is therefore deformed with respect to the amplitude distribution that would be observed if the angle of incidence were zero. It should be noted that we are considering exclusively the case of a simple reflection produced by the parabolic mirror; the case of multiple reflections is beyond the scope of this paper (for multiple reflections, see for example Ref. [16

16. M. Avendaño-Alejo, L. Castañeda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78(11), 1195–1198 (2010). [CrossRef]

].).

Lieb and Meixner pointed out that the parabolic mirror shows strong coma if it is not illuminated exactly along the mirror axis [5

5. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]

]. Thus the alignment of the propagation axis of the incident beam is critical in order to use a parabolic mirror as a focusing device. To calculate the electric field of a focused beam in the focal region of a slightly tilted parabolic mirror, Lieb and Meixner included an additional phase factor into the integral representation of the focused electric field, in such a way as to take into account the tilted wavefront. Nonetheless, they did not provide the explicit form of the phase factor.

The electromagnetic field distribution in the focal region of an aberration-free high numerical aperture parabolic mirror has been investigated by a few authors [17

17. C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 1(4), 129–132 (1977). [CrossRef]

19

19. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000). [CrossRef]

]. However, to the best of our knowledge, a detailed theoretical study of the focusing properties of a parabolic mirror used with an incident beam having a nonzero angle of incidence is not available in the literature. It is of interest to obtain an explicit expression for the intensity distribution in the focal region of a slightly tilted parabolic mirror, since this may give a clearer understanding of the effect of the angle of incidence on the focusing properties of a parabolic mirror as such a focusing element may be increasingly used in microscopy systems. The aim of this work is to provide a clear insight into the focusing of the lowest-order radially polarized beam with a parabolic mirror when the angle of incidence is nonzero.

This paper is organized as follows. In section 2, we apply the Richards–Wolf theory to analyze the electric field of a focused beam in the focal region of a parabolic mirror. In section 3, we provide the expression of the electric field of a tightly focused TM01 beam having a nonzero angle of incidence on the mirror. In section 4, we use the expression presented in section 3 to investigate in detail the characteristics of the focused TM01 beam.

2. Focusing a collimated beam with a slightly tilted parabolic mirror

The Richards–Wolf theory consists in a vector diffraction integral that can be applied to describe the electromagnetic fields of strongly focused beams [20

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

]. We consider an incident collimated beam, whose electric field has a given state of polarization and amplitude distribution in the entrance pupil of a parabolic mirror of focal length f. At the exit pupil of the mirror, the beam converges toward the focal region. The coordinate system (r,ϕ,z) is attached to the parabolic mirror and its origin is centered at the focal point of the mirror. The Richards–Wolf vector field equation is valid under the assumption of the Debye approximation, according to which the field can be determined as a superposition of plane electromagnetic waves. The Debye approximation is applicable for high numerical aperture systems when the focal length is much larger than the wavelength in vacuum λ of the beam, i.e. kf >> 1, where k = 2π/λ is the wave number of the illumination [21

21. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981). [CrossRef]

]. This condition is always satisfied in optical microscopy and astronomy. Explicitly, the electric field E(r,ϕ,z) in the neighborhood of the focus is an integral over a vector field amplitude [5

5. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]

,8

8. 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).

,22

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

,23

23. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 3.

]:
E(r,ϕ,z)=1j2π02π0αmaxEo(α)exp[jkΓ(α,β)]sinαdαdβ,
(1a)
Eo(α,β)a^(α,β)Eoq(α)l0(α),
(1b)
Γ(α,β)Φ(α,β)rsinαcos(βϕ)+zcosα,
(1c)
where Φ(α,β) is the aberration function [20

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

], Eo is an arbitrary constant amplitude, α and β are respectively the polar and azimuthal angles of the spherical coordinates in the spatial frequency domain. Herein, the time dependence exp(jωt) is omitted. The unit vector a^(α,β) represents the polarization direction of the focused electric field. The function q(α) is the apodization factor of the focusing system, found from energy conservation and geometric considerations. The function l 0(α) is the amplitude distribution of the collimated input beam at the entrance pupil and it is assumed to be axially symmetric, i.e. β independent. The integration in Eq. (1a) is done over the solid angle that subtends the entrance pupil of the optical system (supposed to be without central obscuration throughout this paper), which is covered with the angles 0β2π and 0ααmax, where α max is the maximum value for the polar angle.

In the following, we consider the focusing with a parabolic mirror whose axis of revolution coincides with the Z-axis. For such a focusing system, the aperture angle α max can be greater than π/2 and may approach π when the transverse dimensions of the mirror are very large and the focal length is relatively short. The apodization factor of a parabolic mirror is given by [5

5. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]

,11

11. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004). [CrossRef] [PubMed]

,24

24. J. J. Stamnes, Waves in Focal Regions (Hilger, 1986), Sec. 16.1.2.

,25

25. C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993). [CrossRef]

]
q(α)=21+cosα=sec2(12α),
(2)
while it is well known that the apodization factor of an aplanatic lens is q(α)=cos1/2α [20

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

,23

23. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 3.

].

g(α)=2sinα1+cosα=2tan(12α).
(3)

We now consider an incident collimated beam having an angle of incidence δ with respect to the Z-axis. In the following, we assume that the angle δ is very small so that the small-angle approximation applies, i.e. sinδδ and cosδ1. When the propagation axis of a collimated beam is tilted with a very small angle with respect to the Z-axis in the x-z plane (see Fig. 2), the field distribution acquires an additional phase factor given by exp(jkxδ). The coordinate x can be written in terms of the angular spherical coordinates α, β as x=fg(α)cosβ.

The function Φ(α,β)fδg(α)cosβ can be seen as the aberration function of the optical system. Using Eq. (3), the aberration function of the parabolic mirror is then Φ(α,β)=2fδtan(12α)cosβ. To get some insight into the form of this particular aberration function, let us expand the function 2tan(12α) as a power series of sinα: 2tan(12α)sinα+14sin3α+ for 0απ/2. Hence, the first terms of the aberration function are

Φ(α,β)=2fδtan(12α)cosβfδsinαcosβ+14fδsin3αcosβ+
(4)

The first term corresponds to the distortion, which displaces the position of the focal spot by an amount along the X-axis with respect to the focus of the parabolic mirror [26

26. R. Kant, “Vector diffraction in paraboloidal mirrors with Seidel aberrations. I: Spherical aberration, curvature of field aberration and distortion,” Opt. Commun. 128(4-6), 292–306 (1996). [CrossRef]

,27

27. C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997). [CrossRef]

]. The second term is associated to the primary coma whose aberration coefficient is 14fδ [27

27. C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997). [CrossRef]

,28

28. R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(7), 075008 (2008). [CrossRef]

]. Higher-order terms correspond to higher-order coma such as secondary coma, etc. Therefore, since the focal length f of the mirror is large (many wavelengths), we expect that the tilting of the propagation axis of the collimated beam incident on a parabolic mirror results in an electric energy density affected by severe coma, even for small values of the angle of incidence δ. For a rigorous analysis, one should include in the present treatment the effect of the change in the apodization factor resulting from the presence of aberrations; however, this effect can be neglected because it is assumed that q(α)δ << 1 for 0ααmax.

When a collimated beam is incident on an aplanatic lens with a nonzero angle of incidence, the field distribution in the focal plane does not suffer from coma. In fact, the aberration function of the aplanatic lens is Φ(α,β)fδsinαcosβ, because g(α)=sinα in this case and it corresponds only to distortion. Thus, the shape of the field distribution at the focus of an aplanatic lens remains unchanged by a nonzero angle of incidence, but the field distribution is shifted along the X-axis in the focal plane.

The specific form of the aberration function defined by Eq. (4) can be taken into account in such a way that it does not formally alter the aberration-free field. Substituting the aberration function Φ(α,β)=fδg(α)cosβ (where g(α) is given by Eq. (3) for the parabolic mirror) in the argument of the exponential function defined by Eq. (1c) yields

Γ(α,β)=sinα[rcosϕcosβfδg(α)sinαcosβ+rsinϕsinβ]+zcosα.
(5)

The trigonometric identity acosβ+bsinβ=(a2+b2)1/2cos(βθ) where tanθ=b/a can be used to rewrite the expression between the brackets in Eq. (5):

Γ(α,β)=r˜(α)sinαcos[βϕ˜(α)]+zcosα,
(6a)
r˜(α)[r2+f2δ2g2(α)sin2α2rfδg(α)sinαcosϕ]1/2,
(6b)
tanϕ˜(α)rsinϕrcosϕfδg(α)/sinα.
(6c)

The function Γ(α,β) contains all the dependence over the azimuthal angle β. In light of Eq. (6a), the function Γ(α,β) has the same form as in the case of an aberration-free electric field. In other words, Eq. (1c) with Φ=0 is formally identical to Eq. (6a), provided that the substitutions rr˜(α) and ϕϕ˜(α) are done.

3. Focusing a TM01 beam with a slightly tilted parabolic mirror

We now apply the Richards–Wolf theory to the special case of a slightly tilted TM01 beam focused with a parabolic mirror. The TM01 beam is a transverse magnetic (TM) beam and it is the lowest-order member of the family of the radially polarized beams. The TM01 beam in particular is analyzed in detail because of its practical importance compared to other transverse magnetic, transverse electric or linearly polarized beams. As mentioned before, Quabis et al. showed that strong focusing of radially polarized light leads to tighter spot sizes than is possible with linearly polarized light [7

7. 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]

]. Because of their remarkable focusing properties, TM01 beams are of substantial interest, for example, in high-resolution microscopy. Furthermore, the electric field of a strongly focused TM01 beam has a significant longitudinal component that can be used in particle trapping and electron acceleration [29

29. C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2 ), 026603 (2005). [CrossRef] [PubMed]

].

Many methods have been demonstrated to generate TM01 beams in the laboratory. To name a few, a TM01 beam can be produced interferometrically, outside the laser cavity, with a Mach–Zehnder interferometer which allows the coherent superposition of two orthogonally polarized Laguerre–Gaussian beams of order (0,1) of different parity with the same beam waist [30

30. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]

]. A technique to produce pseudo-radially polarized beams requires a polarization converter consisting in four half-wave plates, one in each quadrant [9

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

]. Besides, radially polarized beams may be generated directly from a laser by inserting axially-symmetric optical elements with suitable polarization selectivity in the laser cavity; such elements include a conical reflector used as a resonator mirror, a conical Brewster window and a birefringent c-cut laser crystal [31

31. H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008). [CrossRef] [PubMed]

].

Lieb and Meixner presented electric energy densities in the focal region of a parabolic mirror in the case of a TM01 beam focused with an angle of incidence whose value approximately lies between 0° and 0.016° [5

5. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]

]. Their calculated electric energy density distributions exhibit asymmetry that is characteristic of coma. In this section, we show a detailed approach to obtain such results. Since the maximum acceptable value for the angle of incidence δ is close to 0.02° for practical applications requiring high resolution (as it will be apparent in Section 4), it is reasonable to consider that the amplitude l 0(α) of the field distribution and the unit vector a^(α,β) defining the state of polarization of the incident beam are not significantly affected by the angle of incidence, i.e. they do not depend on δ. In fact, the vector amplitude distribution of the collimated beam at the entrance pupil is not as sensitive to a nonzero angle of incidence as its phase is. If the effect of the angle of incidence δ on the vector amplitude of the collimated input beam has to be taken into account, one may follow the approach given for example in Ref. [32

32. R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995). [CrossRef]

].

For an incident radially polarized beam, the unit vector a^(α,β) is given by [8

8. 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).

,22

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

]
a^(α,β)=a^xcosαcosβ+a^ycosαsinβ+a^zsinα,
(7)
where a^x,a^y,a^z are the unit vectors oriented along the Cartesian axes x,y,z attached to the parabolic mirror and whose origin is positioned at the focus of the mirror. Using Eqs. (6a) and (7), Eq. (1) can be rewritten explicitly as
{ExEyEz}=Eoj2π02π0αmaxq(α)l0(α){cosαcosβcosαsinβsinα}×exp{jkr˜(α)sinαcos[βϕ˜(α)]}exp(jkzcosα)sinαdαdβ.
(8)
The integration over β can be carried out using the following identities [8

8. 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).

,20

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

]:
02πexp[jkrsinαcos(βϕ)]{cos(mβ)sin(mβ)}dβ=2πjmJm(krsinα){cos(mϕ)sin(mϕ)},
(9)
where Jm() is the Bessel function of the first kind of order m. Using Eq. (9) in Eq. (8), the electric field near focus becomes
{ExEyEz}=Eo0αmaxq(α)l0(α)sinα{cosαcosϕ˜(α)J1[kr˜(α)sinα]cosαsinϕ˜(α)J1[kr˜(α)sinα]jsinαJ0[kr˜(α)sinα]}exp(jkzcosα)dα,
(10)
where, with the help of Eq. (6c), the cosine and sine of angle ϕ˜(α) can be expressed as

cosϕ˜(α)=rcosϕfδg(α)/sinαr˜(α),
(11a)
sinϕ˜(α)=rsinϕr˜(α).
(11b)

Finally, the amplitude distribution of the collimated paraxial TM01 beam incident on the parabolic mirror reads
l0(α)=rwoexp(r2wo2)=fg(α)woexp[f2g2(α)wo2],
(12)
where the relation r=fg(α) has been used and where wo is the spot size of the incident beam at its waist. Solving Eq. (10) for the apodization factor q(α) defined by Eq. (2) and with the field amplitude l 0(α) and the function g(α) given by Eqs. (12) and (3), respectively, leads to the electric field vector E(r,ϕ,z) of a the TM01 beam focused with a slightly tilted parabolic mirror. The electric energy density We in the focal region is proportional to the square modulus of the focused electric field of the beam, i.e. We12ε0|E|2, where ε0 is the permittivity of free space. It should be noted that focused field has the following symmetries: Ex(r,ϕ,z)=Ex(r,ϕ,z), Ey(r,ϕ,z)=Ey(r,ϕ,z), and Ez(r,ϕ,z)=Ez(r,ϕ,z), so that We(r,ϕ,z)=We(r,ϕ,z).

4. Numerical simulations

We present in this section some profiles of the electric energy density, which is defined by We12ε0|E|2=12ε0(|Ex|2+|Ey|2+|Ez|2). Unless otherwise stated, all the results are normalized with respect to the maximum intensity of the focused beam for which ka = 1 and δ = 0. For a given value of ka, increasing the value of kfδ deforms the shape of the electric energy density profile. For ka = 1, as kfδ increases from zero to 3π, the position of the peak intensity is shifted and the rotational symmetry of the intensity distribution in the x-y plane is converted into a uniaxial symmetry with respect to the X-axis (Fig. 3
Fig. 3 The electric energy density distribution We in the focal plane (z = 0) of a focused TM01 beam by a parabolic mirror for which ka = 1 with kfδ equal to (a) 0, (b) 1.5π, and (c) 3π.
). With a zero angle of incidence, the structure of the electric energy density is aberration free and it possesses lobes of relatively weak amplitude surrounding a central high amplitude peak [Fig. 3(a)]. When kfδ is different from zero, the rotational symmetry exhibited in the intensity distribution of a beam focused with a zero angle of incidence is destroyed by formation of unsymmetrical rings, showing a comatic image flaring in the X-direction [Figs. 3(b) and 3(c)]; these distributions, characterized by a comet-like shape, confirm the presence of coma in the focal spot when the beam is incident on the parabolic mirror with an angle of incidence, however small it is. Interestingly, the aberrated beam can be seen as a superposition of the nonparaxial fundamental TM01 beam and higher-order TM beams [4

4. A. April and M. Piché, “4π Focusing of TM(01) beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010). [CrossRef] [PubMed]

]. The position of the best focus, its full-width at half-maximum (FWHM), and the value of the maximum intensity are affected by a change in the value of kfδ.

The intensity profile of a focused TM01 beam in the focal plane of a parabolic mirror is strongly dependent on beam divergence angle, i.e. on parameter ka. Figure 4
Fig. 4 Electric energy density profiles along the X-axis (y = 0) in the focal plane (z = 0) for selected values of ka and kfδ. Note the different scale for each plot.
shows the profiles of the electric energy density for different values of kfδ, for ka = 1, 4, and 7 (which correspond to beam divergence angles approximately equal to 66°, 40°, and 30°, respectively). Each profile is on the X-axis (y = 0), so that Ey = 0. As kfδ increases, an extended tail develops in the electric energy density profile, especially for low values of ka. The presence of comatic aberration in the system yields a positional displacement and a modification of the width of the central peak intensity accompanied with, in most cases, a reduction of the peak intensity.

Due to its high amplitude, the longitudinal component of the electric field plays a dominating role in shaping the total electric energy density for ka < 7 (in the nonparaxial regime). However, for ka = 7, the transverse components of the electric field have nearly the same magnitude as the longitudinal component. For kfδ = 0, the transverse components of the electric field are zero at the focal point, providing a dark core to the electric energy density profile associated to the transverse components, as shown in the first row of Fig. 5
Fig. 5 Electric energy density distribution along the X-axis (y = 0) in the focal plane (z = 0) of a focused TM01 beam for which for ka = 7, for kfδ = 0 and kfδ = 6π. (a) |Ex|2, (b) |Ez|2, and (c) |Ex|2+|Ez|2. The profiles are normalized by the maximum intensity of |Ex|2+|Ez|2.
. As kfδ increases, the energy in the lobe closer to the origin is gradually transferred to the lobe farther from the origin, enhancing the peak intensity of the latter, as illustrated in the second row of Fig. 5. Therefore, the peak intensity of the electric energy density distribution associated to the total field can be higher than the aberration-free case, due to this redistribution of energy in the focal plane from one peak to the adjacent. Note that this effect is observed in Fig. 4 for the case of ka = 7.

The Strehl ratio is the ratio of the observed peak intensity in the focal plane (z = 0) of the parabolic mirror compared to the maximum peak intensity of the aberration-free version of the system, i.e. when δ = 0. Specifically, we define the Strehl ratio by SI/I0, where I 0 is the maximum value of the electric energy density We of the unaberrated beam and I is the maximum value of the electric energy density of the aberrated beam taken at the plane z = 0. For ka = 1, the Strehl ratio decreases relatively quickly with an increase in the value of kfδ; for ka = 4, it decreases, but more slowly; and for ka = 7, it is greater than one, at least for sufficient low values of kfδ [Fig. 6(a)
Fig. 6 (a) The Strehl ratio, (b) the full-width at half-maximum (FWHM), and (c) the position x max of the peak intensity of the electric energy density profile in the focal plane (z = 0) as a function of the parameter kfδ, for ka = 1 (red curves), for ka = 4 (green curves), and for ka = 7 (blue curves).
]. This last observation can be explained by the fact that the amplitude of the transverse electric field components is high enough to shape the total electric energy density in such a way that the energy transferred from one lobe to the other increases the maximum value of the peak intensity (see Fig. 5).

The FWHM of the central lobe, calculated on the X-axis (y = 0), is affected by the angle of incidence of the beam on the parabolic mirror [Fig. 6(b)]. It increases with an increase in the value of kfδ for ka = 1; it is roughly constant for ka = 4; and it decreases with kfδ for ka = 7. The last observation can be understood by the fact that the total electric energy density profile is, for δ = 0, the superposition of three juxtaposed peaks of nearly identical height, whereas it is the superposition of practically two peaks when kfδ is larger, since the amplitude of the lobe nearer to the origin becomes relatively small.

With a nonzero value of kfδ, the electric energy density profile is shifted in one direction with respect to the origin. The position of the peak of highest intensity, denoted by x max, depends on the amount of aberration, but it is almost independent of the beam divergence angle [Fig. 6(c)]. According to the results of Fig. 6(c), the position of the best focus in the plane z = 0 is approximately located at x max = 1.3, measured along the X-axis with respect to the focal point of the paraboloid (this is valid at least for 1ka7 and 0kfδ6π). This displacement is mainly explained by the presence of distortion in the aberration function of the system; the extra amount of displacement can be attributed to the presence of coma in the aberration function of the system.

5. Conclusion

In summary, we outlined a method to analyze quantitatively the effects of the angle of incidence on the focusing properties of a high numerical aperture parabolic mirror. The aberration function of the system, for a very small tilting of the mirror, can be expanded to underline the fact that the system suffers from distortion and coma. The application of the Richards–Wolf theory to the tight focusing of a TM01 beam yields the electric field of the strongly focused beam in the focal region. This analysis shows that, when a parabolic mirror is illuminated with an incident collimated beam whose axis of propagation does not exactly coincide with the axis of revolution of the mirror, a positional displacement of the focal spot, a change in the width of the central lobe and a modification of the peak intensity are observed.

The analysis presented in this paper is consistent with the results of Lieb and Meixner, which claim that, for kf = 18,000π, an angle of incidence as small as 0.006° causes a severe asymmetry of the electric energy density, resulting from the presence of coma [5

5. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]

]. Consequently the propagation axis of the incident beam must be properly aligned with respect to the axis of revolution of the mirror axis. The correction of the misalignment can be achieved by a steering mirror equipped with micrometer adjustment screws [5

5. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]

]. Other techniques to correct the aberrations in a parabolic mirror involve closed loop adaptive systems [34

34. J. Sheldakova, A. Kudryashov, and T. Y. Cherezova, “Femtosecond laser beam improvement: correction of parabolic mirror aberrations by means of adaptive optics,” Proc. SPIE 6872, 687203 (2008). [CrossRef]

].

Acknowledgments

The authors acknowledge support from Natural Sciences and Engineering Research Council of Canada (NSERC), Fonds québécois de recherche sur la nature et les technologies (FQRNT), Canadian Institute for Photonic Innovations (ICIP/CIPI), and the Centre d'optique, photonique et laser (COPL), Québec.

References and links

1.

H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM(01) laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009). [CrossRef] [PubMed]

2.

H. Ichikawa and H. Kikuta, “Numerical feasibility study of the fabrication of subwavelength structure by mask lithography,” J. Opt. Soc. Am. A 18(5), 1093–1100 (2001). [CrossRef]

3.

N. Bokor and N. Davidson, “4π focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008). [CrossRef]

4.

A. April and M. Piché, “4π Focusing of TM(01) beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010). [CrossRef] [PubMed]

5.

M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]

6.

A. Drechsler, M. A. Lieb, C. Debus, A. J. Meixner, and G. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9(12), 637–644 (2001). [CrossRef] [PubMed]

7.

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]

8.

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).

9.

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

10.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007). [CrossRef]

11.

N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004). [CrossRef] [PubMed]

12.

J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008). [CrossRef] [PubMed]

13.

C. Debus, M. A. Lieb, A. Drechsler, and A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003). [CrossRef] [PubMed]

14.

E. W. Weisstein, “Catacaustic.” From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/Catacaustic.html.

15.

J. E. Howard, “Imaging properties of off-axis parabolic mirrors,” Appl. Opt. 18(15), 2714–2722 (1979). [CrossRef] [PubMed]

16.

M. Avendaño-Alejo, L. Castañeda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78(11), 1195–1198 (2010). [CrossRef]

17.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 1(4), 129–132 (1977). [CrossRef]

18.

R. Barakat, “Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration,” Appl. Opt. 26(18), 3790–3795 (1987). [CrossRef] [PubMed]

19.

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000). [CrossRef]

20.

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

21.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981). [CrossRef]

22.

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

23.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 3.

24.

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986), Sec. 16.1.2.

25.

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993). [CrossRef]

26.

R. Kant, “Vector diffraction in paraboloidal mirrors with Seidel aberrations. I: Spherical aberration, curvature of field aberration and distortion,” Opt. Commun. 128(4-6), 292–306 (1996). [CrossRef]

27.

C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997). [CrossRef]

28.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(7), 075008 (2008). [CrossRef]

29.

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2 ), 026603 (2005). [CrossRef] [PubMed]

30.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]

31.

H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008). [CrossRef] [PubMed]

32.

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995). [CrossRef]

33.

G. Rodríguez-Morales and S. Chávez-Cerda, “Exact nonparaxial beams of the scalar Helmholtz equation,” Opt. Lett. 29(5), 430–432 (2004). [CrossRef] [PubMed]

34.

J. Sheldakova, A. Kudryashov, and T. Y. Cherezova, “Femtosecond laser beam improvement: correction of parabolic mirror aberrations by means of adaptive optics,” Proc. SPIE 6872, 687203 (2008). [CrossRef]

OCIS Codes
(080.1010) Geometric optics : Aberrations (global)
(230.4040) Optical devices : Mirrors
(260.1960) Physical optics : Diffraction theory
(260.5430) Physical optics : Polarization
(350.5500) Other areas of optics : Propagation
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Physical Optics

History
Original Manuscript: March 8, 2011
Revised Manuscript: April 17, 2011
Manuscript Accepted: April 18, 2011
Published: April 26, 2011

Virtual Issues
Vol. 6, Iss. 6 Virtual Journal for Biomedical Optics

Citation
Alexandre April, Pierrick Bilodeau, and Michel Piché, "Focusing a TM01 beam with a slightly tilted parabolic mirror," Opt. Express 19, 9201-9212 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9201


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References

  1. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM(01) laser beam at a dielectric interface,” Opt. Lett. 34(23), 3601–3603 (2009). [CrossRef] [PubMed]
  2. H. Ichikawa and H. Kikuta, “Numerical feasibility study of the fabrication of subwavelength structure by mask lithography,” J. Opt. Soc. Am. A 18(5), 1093–1100 (2001). [CrossRef]
  3. N. Bokor and N. Davidson, “4π focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008). [CrossRef]
  4. A. April and M. Piché, “4π Focusing of TM(01) beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010). [CrossRef] [PubMed]
  5. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef] [PubMed]
  6. A. Drechsler, M. A. Lieb, C. Debus, A. J. Meixner, and G. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9(12), 637–644 (2001). [CrossRef] [PubMed]
  7. 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]
  8. 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).
  9. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
  10. T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007). [CrossRef]
  11. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004). [CrossRef] [PubMed]
  12. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008). [CrossRef] [PubMed]
  13. C. Debus, M. A. Lieb, A. Drechsler, and A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003). [CrossRef] [PubMed]
  14. E. W. Weisstein, “Catacaustic.” From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/Catacaustic.html .
  15. J. E. Howard, “Imaging properties of off-axis parabolic mirrors,” Appl. Opt. 18(15), 2714–2722 (1979). [CrossRef] [PubMed]
  16. M. Avendaño-Alejo, L. Castañeda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78(11), 1195–1198 (2010). [CrossRef]
  17. C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 1(4), 129–132 (1977). [CrossRef]
  18. R. Barakat, “Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration,” Appl. Opt. 26(18), 3790–3795 (1987). [CrossRef] [PubMed]
  19. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000). [CrossRef]
  20. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959). [CrossRef]
  21. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981). [CrossRef]
  22. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]
  23. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 3.
  24. J. J. Stamnes, Waves in Focal Regions (Hilger, 1986), Sec. 16.1.2.
  25. C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993). [CrossRef]
  26. R. Kant, “Vector diffraction in paraboloidal mirrors with Seidel aberrations. I: Spherical aberration, curvature of field aberration and distortion,” Opt. Commun. 128(4-6), 292–306 (1996). [CrossRef]
  27. C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997). [CrossRef]
  28. R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(7), 075008 (2008). [CrossRef]
  29. C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2 ), 026603 (2005). [CrossRef] [PubMed]
  30. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]
  31. H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008). [CrossRef] [PubMed]
  32. R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995). [CrossRef]
  33. G. Rodríguez-Morales and S. Chávez-Cerda, “Exact nonparaxial beams of the scalar Helmholtz equation,” Opt. Lett. 29(5), 430–432 (2004). [CrossRef] [PubMed]
  34. J. Sheldakova, A. Kudryashov, and T. Y. Cherezova, “Femtosecond laser beam improvement: correction of parabolic mirror aberrations by means of adaptive optics,” Proc. SPIE 6872, 687203 (2008). [CrossRef]

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