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. 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]

Am. J. Phys.

• 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]

Appl. Opt.

• 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]
• S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]
• J. E. Howard, “Imaging properties of off-axis parabolic mirrors,” Appl. Opt. 18(15), 2714–2722 (1979). [CrossRef] [PubMed]

Appl. Phys. B

• 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).

IEE J. Microwaves, Opt. Acoust.

• 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]

J. Microsc.

• 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]

J. Mod. Opt.

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

J. Opt. A, Pure Appl. Opt.

• 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]

J. Opt. Soc. Am. A

• 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]
• 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]

Opt. Commun.

• N. Bokor and N. Davidson, “4π focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008). [CrossRef]
• 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]
• T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007). [CrossRef]
• 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]
• 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]
• 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]

Opt. Express

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

Opt. Lett.

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

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

• 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]

Phys. Rev. Lett.

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

Proc. R. Soc. Lond. A Math. Phys. Sci.

• 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]

Proc. SPIE

• 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]

Other

• L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 3.
• J. J. Stamnes, Waves in Focal Regions (Hilger, 1986), Sec. 16.1.2.
• E. W. Weisstein, “Catacaustic.” From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/Catacaustic.html .

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