Editor-in-Chief : V.K. Rastogi
| Asian Journal of Physics | Vol. 34, Nos 7 & 8 (2025) 375-413 |
Bessel beams, propagationally invariant beams, and axicons: An historical and tutorial review
Colin J R Sheppard
Honorary Professor
School of Optometry and Vision Science, UNSW Sydney, NSW 2052, AustraliaDedicated to Prof Kehar Singh on the occasion of his 84th Birthday on July 3, 2025
The Bessel beam is discussed, including its historical development. We present the relationship with the Arago Spot, annular pupils, axicons and pupil filters. We also discuss Bessel-Gauss beams; higher order Bessel beams; electromagnetic polarization properties; waveguide and resonator modes; polychromatic and pulsed beams; one-dimensional Airy beams; extended depth of focus; and light sheets and photonic nanojets. © Anita Publications. All rights reserved.
Doi: XXX
Keywords: Bessel beams, Propagationally invariant beams, Diffraction-free beams, Nondiffracting beams.
Peer Review Information
Method: Single- anonymous; Screened for Plagiarism? Yes
Buy this Article in Print © Anita Publications. All rights reserve
References
- Sheppard CJR, Wilson T, Imaging properties of annular lenses, App Opt, 18(1979)3764–3769.
- Sheppard CJR, The use of lenses with annular aperture in scanning optical microscopy, Optik, 48(1977)329–334.
- Sheppard CJR, Choudhury A, Image formation in the scanning microscope, Optica Acta, 24(1977)1051–1073.
- Sheppard CJR, Electromagnetic field in the focal region of wide-angular annular lens and mirror systems, IEE J Microwaves, Optics and Acoustics, 2(1978)163–166.
- Sheppard CJR, Wilson T, Gaussian-beam theory of lenses with annular aperture, IEE J Microwaves, Optics and Acoustics, 2(1978)105–112.
- Linfoot E H, Wolf E, Diffraction images in systems with an annular aperture, Proc Phys. Soc B, 66(1953)145–149.
- Sheppard CJR, Scanning microscopes, United States Patent No. 4,198,571, (1980).
- Airy G B, On the diffraction of an annular aperture, Phil Mag Ser, 18(1841)1–10.
- Rayleigh JWS, On the diffraction of object glasses, Monthly Notes of the Royal Astronomical Society, 33(1872)59–63.
- McLeod J H, The axicon: A new type of optical element, J Opt Soc Am, 44(1954)592–597.
- McLeod J H, Axicons and their uses, J Opt Soc Am, 50(1960)166–169.
- Dyson J, Circular and spiral diffraction gratings. Proc Royal Soc London Ser A, 248(1958)93–106.
- Vasara A, Turunen J, Friberg AT, Realization of general nondiffracting beams with computer-generated holograms, J Opt Soc Am A, 6(1989)1748–1754.
- Rosen J, Yariv A, Synthesis of an arbitrary axial field profile by computer-generated holograms, Opt Lett, 19(1994)843–845.
- Thompson B J, Diffraction by semitransparent and phase annuli, J Opt Soc Am, 55(1965)145–149.
- Wang H F, Gan F, High focal depth with a pure-phase apodizer, App Opt, 40(2001)5658–5662.
- Potter R, On the Phenomena of diffraction in the centre of the shadow of a circular disc. Phil Mag, 19(1841)153–154.
- Lommel E, Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet, Abh Bayer Akad, 15(1885)233–328.
- Porter A W, On the formation of images by means of an opaque disk, Phil Mag, 27(1914)673–674.
- Rankine A O, Demonstration on the diffraction of light by a spherical object, Proc Phys Soc, 37(1925)267–267.
- Raman C V, Krishnan K S, On the diffraction of light by spherical objects, Proc Phys Soc, 38(1926)350–353.
- Berko S, Lee Y G, Wright F (Jr), Rosenfeld J, Undergraduate laser diffraction experiments using large objects, Am J Phys, 38(1970)348–352.
- Harvey J E, Forgham J L, The spot of Arago: New relevance for an old phenomenon. Am J Phys, 52(1984)243–247.
- Harrison M E, Marek C T, White J D, Rediscovering Poisson’s spot, The Physics Teacher, 35(1997)18–19.
- Saari P, Bowlan P, Valtla-Lukner H, Löhmus M, Piksarv P, Trebino R, Basic diffraction phenomena in time domain, Opt Exp, 18(2010)11083–11088.
- Airy G B, On the diffraction of an object-glass with circular aperture, Trans Cambridge Phil Soc, 5(1835)283–291.
- Rayleigh J W S, Wave Theory of Light, Encyclopaedia Britannica, Adam and Charles Black: Edinburgh, (1888) 421–459.
- Steward G C, IV Aberration diffraction effects, Phil Trans Royal Soc London Ser A, 225(1926)131–198.
- Steward G C, The Symmetrical Optical System, (Cambridge University Press, Cambridge, U K, (1928).
- Born M, Wolf E, Principles of Optics, 1st edn, (Oxford: Pergamon), 1959.
- Dunham T, The effect of central stops on the resolution of a microscope, Thesis, MIT: Boston MA, 1948.
- Steel W H, Etude des effets combinés des aberrations et d’une obturation centrale de la pupille sur le contraste des images optiques, Rev d’Optique, 32(1953)4.
- O’Neill E L, Transfer function for an annular aperture, J Opt Soc Am, 46(1956)285–288.
- Riesenberg H, Annular apertures, Jenaer Jahrbuch, (1956)30–78.
- Taylor CA, Thompson BJ, Attempt to investigate experimentally the intensity distribution near the focus in error free diffraction patterns of circular and annular aperture, J Opt Soc Am, 48(1958)844–850.
- Welford W T, Use of annular apertures to increase focal depth, J Opt Soc Am, 50(1960)749–753.
- Barakat R, Total illumination in a diffraction image containing spherical aberration, J Opt Soc Am, 51(1961)152–157.
- Kelly D H, Axicons, annuli, and coherent light, J Opt Soc Am, 53(1963)525.
- Mahan A I, Bitterei C V, Cannon S M, Far-field diffraction patterns of single and multiple apertures bounded by arcs and radii of concentric circles, J Opt Soc Am, 54(1964)721–732.
- Izatt J R, Diffraction patterns produced by apertures bounded by arcs and radii of concentric circles, J Opt Soc Am, 55(1965)106–107.
- Barakat R, Houston A, Transfer funtion of an annular aperture in the presence of spherical aberration, J Opt Soc Am, 55(1965)538–541.
- Mehta B L, Total illumination an aberation free annular aperture, App Opt, 13(1974)736–737.
- Tschunko HFA, Irradiance distribution in the error free diffraction image, J Opt Soc Am, 55(1965)1–4.
- Tschunko HFA, Imaging performance of annular apertures, App Opt, 13(1974)1820–1823.
- Tschunko HFA, Annular apertures with low and high obstruction, App Opt, 20(1981)168–169.
- Wiener M M, Useful beam quality design curves for unstable resonators, Opt Eng, 13(1974)87–91.
- Goldberg I L, McCullough A W, Annular aperture diffracted energy distribution for a extended source. App Opt, 8(1969)1451–1458.
- Otterman J, Diffraction-limited resolution for geoscene imagery, App Opt, 8(1969)1887–1889. .
- Stamnes J J, Heier H, Ljunggren S, Encircled energy for systems with centrally obscured circular pupils, App Opt, 21(1982)1628–1633.
- Sodha M S, Aggarwal A K, Sirohi R S, The fluctuations of intensity at the focus of an annular aperture, Optica Acta, 17(1970)623–629.
- Singh K, Kavathekar A K, Image of a general periodic bar pattern through an aberation-free annular aperture, J Opt Soc Am, 59(1969)936–941.
- Dhillon H S, Diffraction of partially coherent light by an annular aperture, Ph D Thesis, IIT Delhi, India, 1968.
- McCrickerd J T, Coherent processing and depth of focus of annular aperture imagery, App Opt, 10(1971)2226–2230.
- Sirohi R S, Bhatnagar G S, Effect of partial coherence on the resolution of a microscope, Optica Acta, 17(1970) 839–842.
- De M, Basuray A, Two point-resolution in partially coherent light: II. Ordinary microscope, annular source, Optica Acta, 19(1972)523–531.
- McKechnie T S, The effect of condenser obstruction on the two point resolution of a microscope, Optica Acta, 19(1972)729–737.
- Nayyar V P, Verma N K, Resolution of a microscope: Improvement using a centrally obsmuted condenser aperture and a semi-transparent π-phase annular aperture objective system, App Opt, 15(1976)1341–1344.
- Rose H, Phase contrast in scanning transmission electron microscopy, Optik, 39(1974)416–436.
- Fenneman D B, Cruise D R, Far-field illumination of targets by annular aperture, J Opt Soc Am, 65(1975)1300–1305.
- Nayyar V P, Verma N K, Diffraction by truncated-Gaussian annular apertures, Optics (Paris), 9(1978)307–310.
- Cox I J, Sheppard C J R, Wilson T, Super-resolution by confocal fluorescent microscopy, Optik, 60(1982)391–396.
- Mahajan V N, Zernike annular polynomials for imaging systems with annular pupils, J Opt Soc Am, 71(1981)75–85.
- Mahajan V N, Zernike annular polynomials for imaging systems with annular pupils. J Opt Soc Am A, 1(1984)685–685.
- Cornacchio J V, Transfer function of an elliptical annular aperture, J Opt Soc Am, 57(1967)1325–1330.
- Bradley L C, Diffraction due to an elliptic annulus, J Opt Soc Am, 66(1976)190.
- Mondal P K, Fraunhofer diffraction due to an elliptic annulus, J Opt Soc Am, 65(1975)1154–1156.
- Tanaka T, Taguchi M, Tanaka K, Fraunhofer diffraction of a plane wave by an elliptical annular aperture with offset or tilt, Opt Commun, 90(1992)207–212.
- Fink D, Polarization effects of axicons, App Opt, 18(1979)581–582.
- Rayces J L, Formation of axicon images, J Opt Soc Am, 48(1958)576–577.
- Steel W H, Axicons with spherical surfaces, in Optics in Metrology, Mollet P, (Ed), Pergamon, (1960), pp 181–193.
- Ronchi V, Das Okularinterferometer und das Objektivinterferometer bei der Auflösung der Doppelsterne. Z Physik, 37(1926)732–757.
- Fujiwara J, Optical properties of conic surfaces, I. Reflecting cone, J Opt Soc Am, 52(1962)287–292.
- Bickel W S, The spectroscopic properties of the ionic beam source, App Opt, 6(1967)1309–1317.
- Bélanger P-A, Rioux M, Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam, App Opt, 17(1978)1080–1086. .
- Pérez M V, Gómez-Reino C, Cuadrado J M, Diffraction patterns and zone plates produced by thin linear axicons, Optica Acta, 33(1986)1161–1176.
- Bakken G S, Parabolic axicon, App Opt, 13(1974)1291–1292.
- Vargady L O, The parabolic axicon: comments, App Opt, 14(1975)2335–2335.
- Bakken G S, Parabolic axicon: reply to comment, App Opt, 14(1975)2335_1-2336.
- Arimoto R, Saloma C, Tanaka T, Kawata S, Imaging properties of axicon in a scanning optical system, App Opt, 31(1992)6653–6657.
- Botcherby E J, Juskaitis R, Wilson T, Scanning two-photon fluorescence microscopy with extended depth of field, Opt Commun, 268(2006)253–260.
- Durnin J, Exact solutions for nondiffracting beams. I. The scalar theory, J Opt Soc Am A, 4(1987)651–654.
- Durnin J, Miceli J J (Jr), Eberly J H, Diffraction-free beams, Phys Rev Lett, 58(1987)1499–1501.
- Indebetouw G, Nondiffracting optical fields: some remarks on their analysis and synthesis, J Opt Soc Am A, (1989)150–152.
- Pieper R J, Poon T-C, Optical transfer functions for defocused two-pupil systems involving annular geometries. J Mod Opt, 37(1990)2055–2072.
- Herman R M, Wiggins T A, Production and uses of diffractionless beams, J Opt Soc Am A, 8(1991)932–942.
- Nakamura O, Toyoda K, Sidelobe suppression of the point-spread function in annular-pupil optical systems. App Opt, 30(1991)3242–3245.
- Cox A J, D’Anna J, Constant-axial-intensity nondiffracting beam, Opt Lett, 17(1992)232–234.
- Herman R M, Wiggins T A, Apodization of diffractionless beams. App Opt, 31(1992)5913–5915.
- Lü B-D, Zhang B, Cai B-W, Yang C-L, Nondiffracting beams with finite beam width, Chinese Sci Bull, 39(1994)801–805.
- Bouchal Z, Wagner J, Olivik M, Bessel beams in the focal region. Opt Eng, 34(1996)1680–1688.
- Jaroszewicz Z, Axicons: Design and propagation properties, (1997) Warsaw: SPIE Polish Chapter.
- Cox A J, Dibble D C, Nondiffracting beam from a spatially filtered Fabry–Perot resonator, J Opt Soc Am A, 9(1992)282–286.
- Jackson DGA, Gu M, Sheppard CJR, 3-D optical transfer function for circular and annular lenses with spherical aberration and defocus, J Opt Soc Am A, 11(1994)1758–1767.
- Strand T C, Werlich H, Aberration limits for annular Gaussian beams for optical storage, App Opt, 33(1994)3533–3539.
- Kowarz M W, Agarwal G S, Bessel-beam representation for partially coherent fields, J Opt Soc Am A, 12(1995)1324–1330.
- Rosen J, Salik B, Yariv A, Pseudo-nondiffracting beams generated by radial harmonic functions, J Opt Soc Am A, 12(1995)2446–2457.
- Rosen J, Salik B, Yariv A, Pseudo-nondiffracting beams generated by radial harmonic functions – erratum, J Opt Soc Am A, 13(1996)387–387.
- Amidror I, Fourier spectrum of radially periodic images, J Opt Soc Am A, 14(1997)816–826.
- Herman R M, Wiggins T A, Rayleigh range and the M2 factor for Bessel–Gauss beams, App Opt, 37(1998)3398– 3400.
- Ruschin S, Leizer A, Evanescent Bessel beam, J Opt Soc Am A, 15(1998)1139–1143.
- Sheppard C J R, Gu M, Three-dimensional optical transfer function for an annular lens, Opt Commun, 81(1991)276–280. .
- Sheppard C J R, Gu M, Optical sectioning in confocal microscopes with annular pupil, Optik, 86(1991)169–172.
- Sheppard C J R, Gu M, Improvement of axial resolution in confocal microscopy using an annular pupil, Opt Commun, 84(1991)7–13.
- Gu M, Sheppard C J R, Three-dimensional coherent transfer function in reflection-mode confocal microscopy using annular lenses, J Mod Opt, 39(1992)783–793.
- Gu M, Sheppard CJR, Axial resolution in the fibre-optical confocal scanning microscope using annular lenses. Opt Commun, 88(1992)27–32.
- Gu M, Sheppard C J R, Zhou H, Optimization of axial resolution in confocal imaging using annular pupils, Optik, 93(1993)87–90.
- Gu M, Sheppard C J R, Three-dimensional imaging in confocal fluorescent microscopy with annular lenses, J Mod Opt, 38(1991)2247–2263.
- Gu M, Sheppard C J R, Effects of annular pupils on confocal fluorescent imaging, J Mod Opt, 39(1992)1883-1896.
- Gu M, Sheppard C J R, Confocal fluorescent microscopy with a finite-sized circular detector, J Opt Soc Am A, 9(1992)151–153.
- Arimoto R, Kawata S, Laser-scan fluorescence microscope with annular excitation, Optik, 86(1990)7–10.
- Gu M, Sheppard, Three-dimensional coherent transfer functions in confocal imaging with two unequal annular lenses, J Mod Opt, 40(1993)1255–1272.
- Gan X S, Sheppard C J R, Imaging in a confocal microscope with one circular and one annular lens, Opt Commun, 103(1993)254–264.
- Gu M, Tannous T, Sheppard C J R, Effect of numerical aperture, pinhole size and annular pupil on confocal imaging through highly scattering media, Opt Lett, 21(1996)312–314.
- Gan X, Sheppard C J R, Gu M, Effects of Fresnel diffraction on confocal imaging with an annular lens, Bioimaging, 5(1997)153–158.
- Rayleigh J W S, On the passage of electric waves through tubes, or the vibrations of dielectric cylinders, Phil Mag, 43(1897)125–132.
- Stratton J, Electromagnetic Theory, (New York: McGraw-Hill), 1941.
- Heckenberg N R, McDuff R, Smith C P, White A G, Generation of optical-phase singularities by computer generated holograms, Opt Lett, 17(1992)221–223.
- Gori F, Guatteri G, Padovani C, Bessel-Gauss beams, Opt Commun, 64(1987)491–495.
- Youngworth K S, Brown T G, Focusing of high numerical aperture cylindrical-vector beams, Opt Exp, 7(2000)77–87.
- Overfelt P K, Bessel-Gauss pulses, Phys Rev A, 44(1991)3941–3947.
- Sheppard CJR, Complex source point theory of paraxial and nonparaxial cosine-Gauss and Bessel–Gauss beams, Opt Lett, 38(2013)564–566.
- Mendoza-Hernandez J, Arroyo-Carrasco M L, Iturbe-Castillo M D, Chavez-Cerda S, Laguerre–Gauss beams versus Bessel beams showdown: peer comparison, Opt Lett, 40(2015)3739–3742.
- Bagini V, Frezza F, Santarsiero M, Schettini G, Shirripa Spagnolo G, Generalized Bessel-Gauss beams, J Mod Opt, 43(1996)1155–1166.
- Herman R M, Wiggins T A, Propagation and focusing of Bessel-Gauss, generalized Bessel-Gauss, and modified Bessel–Gauss beams, J Opt Soc Am A, 18(2001)170–176.
- Sheppard C J R, Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre Gaussian beams, Opt Exp, 17(2009)3690–3697.
- Caron C F R, Potvliege R M, Bessel-modulated Gaussian beams with quadratic radial dependence, Opt Commun, 164(1999)83–93.
- Saghafi S, Sheppard C J R , Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes, J Mod Opt, 45(1998)1999–2009.
- Porras M, Borghi R, Santarsiero M, Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams. J Opt Soc Am A, 18(2001)177–184.
- Sheppard C J R, Porras M A, Comparison between the propagation properties of Bessel–Gauss and generalized Laguerre–Gauss beams, Photonics, 10(2023)1011; doi.org/10.3390/photonics10091011.
- Seshadri S R, Virtual source for the Bessel–Gauss beam, Opt Lett, 27(2002)998–1000.
- Sheppard C J R, High aperture beams, J Opt Soc Am A, 18(2001)1579–1587.
- Hegedus Z S, Annular pupil arrays – application to confocal scanning, Optica Acta, 32(1985)815–826.
- Hegedus Z, Sarafis V, Super-resolving filters in confocally scanned imaging systems, J Opt Soc Am A, 3(1986)1892–1896.
- Sheppard C J R, Hegedus Z S, Axial behavior of pupil plane filters, J Opt Soc Am A, 5(1988)643–647.
- Ojeda-Castañeda J, Gomez-Sarabia C M, Focal depth: optimum annular apodizer, App Opt, 28(1989)4263–4264.
- Fukuda H, Yamanaka R, A new pupil filter for annular illumination in optical lithography, Japan J App Phys, 31(1992)4126–130.
- Sieracki C K, Hansen E W, A leaky annular pupil for improved lateral resolution in confocal fluorescence microscopy, Proc SPIE, 2184(1994)120–126.
- Sheppard C J R, Leaky annular pupils for improved axial imaging, Optik, 99(1995)32–34.
- Sheppard C J R, Synthesis of filters for specified axial properties, J Mod Opt, 43(1996)525–536.
- Xu YS, Singh J, Sheppard CJR, Chen NG, Ultra long high resolution beam by multi-zone rotationally symmetrical complex pupil filter, Opt Exp, 15(2007)6409–6413.
- Cizmar T, Dholakia K, Tunable Bessel light modes: engineering the axial propagation, Opt Exp, 17(2009)15558–15570.
- Wang H F, Gan F, Phase shifting apodizers for increasing focal depth, App Opt, 41(2002)5263–5266.
- Wang H F, Shi L P, Yuan G Q, Miao X S, Tan W L, Chong T C, Subwavelength and super-resolution nondiffraction beam, App Phys Lett, 89(2006)171102; doi.org/10.1063/1.2364693.
- Wang H F, Shi L, Luk`yanchuk B, Sheppard C, Chong C T, Creation of a needle of longitudinally polarized light in vacuum using binary optics, Nature Photonics, 2(2008)501–506.
- Sheppard C J R, Campos J, Escalera J C, Ledesma S, Two-zone pupil filters, Opt Commun, 281(2008)913–922.
- Sheppard C J R, Campos J, Escalera J C, Ledesma S, Three-zone pupil filters, Opt Commun, 281(2008)3623–3630.
- Sheppard C J R, Binary phase filters with maximally-flat response, Opt Lett, 36(2011)1386-1388.
- Sheppard C J R, Mehta S, Three-level filter for increased depth of focus and Bessel beam generation, Opt Exp, 20(2012)27212–27221.
- Rayleigh J W S, On the passage of waves through apertures in plane screens, and allied problems, Phil Mag, 43(1897)259–272.
- Yoshida A, Asakura T, Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular- aperture system, Optik, 40(1974)322–331.
- Shimoda K, Proposal for an electron accelerator using an optical maser, App Opt, 1(1962)33–35.
- Fontana J P, Pantel R H, A high-energy, laser accelerator for electrons using the inverse Cherenkov effect, J App Phys, 54(1983)4285–4288.
- Romea R D, Kimura W D, Modeling of inverse Cherenkov laser acceleration with axicon laser-beam focusing, Phys Rev D, 42(1990)1807–1818.
- Brittingham J N, Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode, J App Phys, 54(1983)1179–1189.
- Mishra S R, A vector wave analysis of a Bessel beam, Opt Commun, 85(1991)159–161.
- Hall D G, Vector-beam solutions of Maxwell’s wave equation, Opt Lett, 21(1996)9–11.
- Jordan R H, Hall D G, Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution, Opt Lett, 19(1994)427–429.
- Greene P L, Hall D G, Diffraction characteristics of the azimuthal Bessel-Gauss beam, J Opt Soc Am A, 13(1996)962–966.
- Greene P L, Hall D G, Properties and diffraction of the azimuthal Bessel-Gauss beam, J Opt Soc Am A, 15(1998)3020–3027.
- Youngworth KS, Brown TG, Inhomogeneous polarization in scanning optical microscopy, Proc SPIE, 3919(2000)75–85.
- Dorn R, Quabis S, Leuchs G, Sharper focus for a radially polarized light beam, Phys Rev Lett, 91(2003)233901; doi.org/10.1103/PhysRevLett.91.233901. .
- Sheppard C J R, Choudhury A, Annular pupils, radial polarization, and superresolution, App Opt, 43(2004)4322–4327.
- Yew E Y S, Sheppard C J R, Tight focusing of radially-polarized Gaussian and Bessel-Gauss beams, Opt Lett, 32(2007)3417–3419.
- Kapany N S, Burke J J, Optical Waveguides, (1972) New York: Academic Press.
- McCutchen C W, Generalized aperture and the three-dimensional diffraction image, J Opt Soc Am, 54(1964)240–244.
- Helseth L E, Optical vortices in focal regions, Opt Commun, 229(2004)85–91.
- Bouchal Z, Olivík M, Non-diffractive vector Bessel beams, J Mod Opt, 42(1995)1555–1566.
- Sick B, Hecht B, Novotny L, Orientational imaging of single molecules by annular illumination, Phys Rev Lett, 85(2000)4482–4485.
- Sheppard C J R, Balla N K, Rehman S, Performance parameters for highly-focused electromagnetic waves, Opt Commun, 282(2009)727–734.
- Sheppard C J R, Balla N K, Rehman S, Yew E Y S, Teng W T, Bessel beams with the tightest focus, Opt Commun, 282(2009)4647–4656.
- Hao X, Kuang CF, Wang TT, Liu X, Phase encoding for sharper focus of azimuthally polarized beam, Opt Lett, 35(2010)3928–3930.
- Sheppard C J R, Rehman S, Highly convergent focusing of light based on rotating dipole polarization, App Opt, 50(2011)4463–4467.
- Sheppard C J R, Polarized focused vortex beams: half-order phase vortices, Opt Exp, 22(2014)18128–18141.
- Lee C S, Lee S W, Chuang S L, Plot of modal field distribution in rectangular and circular waveguides, IEEE Trans Microwave Theory and Techniques, 33(1985)271–274.
- Hondros D, Debye P, Elektromagnetische Wellen an dielektrischen Drähten, Ann Physik, 32(1910)465-476.
- Snitzer E, Cylindrical dielectric waveguide modes, J Opt Soc Am, 51(1961)491–498.
- Kapany NS, Burke JJ, Fiber Optics. IX. Waveguide effects, J Opt Soc Am, 51(1961)1067–1078.
- Gloge D, Weakly guiding fibers, App Opt, 10(1971)2252–2258.
- McCumber DE, Eigenmodes of a symmetric cylindrical confocal laser resonator and their perturbation by output coupling aperture, Bell System Tech J, 44(1965)333–363.
- Nakasiyi M, Ohtsuka V, Annular output coupling for infra-eed gas laser, Japan J App Phys, 6(1967)1305–1310.
- Siegman AE, Arrathoon R, Modes in unstable optical resonators and lens waveguides, IEEE J Quant Elect, 3(1967)156–163.
- Krupke WF, Sooy WR, Properties of an unstable confocal resonator CO2 laser system, IEEE J Quant Elect, 5(1969)575–586.
- Jabczynski J K, A “diffraction-free” resonator, Opt Commun, 77(1990)292–294.
- Pääkkönen P, Turunen J, Resonators with Bessel-Gauss modes, Opt Commun, 156(1998)359–366.
- Rogel-Salazar J, New GHC, Chávez-Cerda S, Bessel-Gauss beam optical resonator, Opt Commun, 190(2003)117–122.
- Sheppard CJR, Gan X, Free-space propagation of femto-second light pulses, Opt Commun, 133(1997)1–6.
- Campbell JA, Soloway S, Generation of a nondiffracting beam with a frequency independent beam width, J Acoust Soc Am, 88(1990)2467–2477.
- Christov I P, Propagation of femtosecond light pulses, Opt Commun, 53(1985)364–366.
- Liu Z Y, Fan D Y, Propagation of pulsed zeroth-order Bessel beams, J Mod Opt, 45(1998)L17–L21.
- Burckhardt C B, Hoffman H, Grandchamp P-A, Ultrasound axicon: a device for focusing over a large depth, J Acoust Soc Am, 54(1973)1628–1630.
- Lu J, Greenleaf J F, Ultrasonic nondiffracting transducer for medical imaging, IEEE Trans Ultrasonics, Ferroelectrics and Frequency Control, 37(1990)438–447.
- Lu J, Greenleaf J F, Nondiffracting X waves: exact solutions to free-space scalar wave equations and their finite aperture realizations, IEEE Trans Ultrasonics, Ferroelectrics and Frequency Control, 39(1992)19–31.
- Fagerholm J, Friberg A T, Huttunen J, Morgan D P, Salomaa M M, Angular-spectrum representation of non-diffracting X waves, Phys Rev E, 54(1996)4347–4352.
- Saari P, Reivelt K, Evidence of X-shaped propagation-invariant localized light waves, Phys Rev Lett, 79(1997)4135–4138.
- Sõnajalg H, Rätsep M, Saari P, Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium, Opt Lett, 22(1997)310–312.
- Recami E, On localized “X-shaped” superluminal solutions to Maxwell equations, Physica A, 252(1998)586–610. 197. Federico A, Martinez O, Distortion of femtosecond pulses due to chromatic aberration in lenses, Opt Commun, 91(1992)104–110.
- Bélanger P A, Packetlike solutions of the homogeneous-wave equation, J Opt Soc Am A, 1(1984)723–724.
- Sezginer A, A general formulation of focus wave modes, J App Phys, 57(1985)678–683.
- Ziolkowski R W, Besieris I M, Shaarawi A M, Aperture realizations of exact solutions to homogeneous-wave equations. J Opt Soc Am A, 10(1993)75–87.
- Heyman E, Pulsed beam propagation in inhomogeneous medium, IEEE Trans Antennas and Propagation, 42(1994)311–319.
- Heyman E, Melamed T, Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation, IEEE Trans Antennas and Propagation, 42(1994)518–525.
- Sheppard CJ R, Bessel pulse beams and focus wave modes, J Opt Soc Am A, 18(2001)2594–2600.
- Reivelt K, Saari P, Bessel–Gauss pulse as an appropriate mathematical model for optically realizable localized waves, Opt Lett, 29(2004)1176–1178.
- Sheppard C J R, Saari P, Lommel pulses: An analytic form for localized waves of the focus wave mode type with bandlimited spectrum, Opt Exp, 16(2008)150–160.
- Sheppard CJR, Generalized Bessel pulse beams, J Opt Soc Am A, 19(2002)2218-2222.
- Sauter T, Paschke F, Can Bessel beams carry superluminal signals? Phys Lett A, 285(2001)1–6.
- Fischer P, Brown CTA, Morris J E, López-Mariscal C, Wright EM, Sibbett W, Dholakia K, White light propagation invariant beams, Opt Exp, 13(2005)6657–6666.
- Wulle T, Herminghaus S, Nonlinear optics of Bessel beams, Phys Rev Lett, 70(1993)1401–1404.
- Glushko B, Kryzhanovsky B, Sarkisyan D, Self-phase-matching mechanism for efficient harmonic generation processes in a ring pump beam geometry, Phys Rev Lett, 71(1993)243; doi.org/10.1103/PhysRevLett.71.243.
- Peatross J, Chaloupka J L, Meyerhofer D D, High-order harmonic generation with an annular laser beam, Opt Lett, 19(1994)942–944.
- Peet V E, Tsubin R V, Third-harmonic generation and multiphoton ionization in Bessel beams, Phys Rev A, 56(1997)1613–1620.
- Tewari S P, Huang H, Boyd R W, Theory of third-harmonic generation using Bessel beams, and self-phase matching, Phys Rev A, 54(1997)2314–2325.
- Hell S W, Hänninen PE, Kuusisto A, Schrader M, Soini E, Annular aperture two-photon excitation microscopy, Opt Commun, 117(1995)20–24.
- Caron C F R, Potvliege RM, Optimum conical angle of a Bessel-Gauss beam for low-order harmonic generation in gases, J Opt Soc Am B, 16(1998)1377–1384.
- Johannisson P, Anderson D, Lisak M, Marklund M, Nonlinear Bessel beams, Opt Commun, 222(2003)107–115.
- Berry M V, Balazs N L, Nonspreading wave packets. Am J Phys, 47(1979)264–267.
- Rosen J, Salik B, Yariv A, Liu H K, Pseudonondiffracting slitlike beam and its analogy to the pseudonondispersing pulse, Opt Lett, 20(1995)423–425.
- Salik B, Rosen J, Yariv A, One-dimensional beam shaping, J Opt Soc Am, 12(1995)1702–1706.
- Ojeda-Castañeda J, Ramos R, Noyola-Isgleas A, High focal depth by apodization and digital restoration, App Opt, 27(1988)2583–2586.
- Sivilogiou GA, Broky J, Dogariu A, Christodoulides DN, Observation of accelerating Airy beams, Phys Rev Lett, 99(2007)213901; doi.org/10.1103/PhysRevLett.99.213901.
- Dowsky E R, Cathey W T, Extended depth of field through wavefront coding, App Opt, 34(1995)1859–1866.
- Arnison M R, Cogswell C J, Sheppard C J R, Török P, Wavefront coding fluorescence microscopy using high aperture lenses, in Optical Imaging and Microscopy, Török P, Kao F-J, Eds, (Springer: Berlin), 2003, pp 143–168.
- Vo S, Fuerschbach K, Thompson K P, Alonso M A, Rolland J P, Airy beams: a geometric optics perspective, J Opt Soc Am A, 27(2010)2574–2582.
- Sheppard C J R, Pupil filters for generation of light sheets, Opt Exp, 21(2013)6339–6345.
- Wilding D, Pozzi P, Soloviev O, Vdovin G, Sheppard C J, Verhaegen M, Pupil filters for extending the field-of view in light-sheet microscopy, Opt Lett, 41(2016)1205–1208.
- Gutiérrez-Vega J C, Iturbe-Castillo M D, Chávez-Cerda S, Alternative formulation for invariant optical fields: Mathieu beams, Opt Lett, 25(2000)1493–1495.
- Ojeda-Castañeda J, Landgrave J E A, Escamilla H M, Annular phase-only mask for high focal depth, Opt Lett, 30(2005)1647–1649.
- Pearcey T, The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic, Phil Mag, 37(1946)311–317.
- Janssen A J E M, On the asymptotics of some Pearcey-type integrals, J Phys A: Math Gen, 25(1992)L823–L831.
- Kirk N P, Connor J N L, Curtis P R, Hobbs C A, Theory of axially symmetric cusped focusing: numerical evaluation of a Bessoid integral by an adaptive contour algorithm, J Phys A: Math Gen, 33(2000)4798–4808.
- Ring J D, Lindberg J, Mourka A, Mazilu M, Dholakia K, Dennis M R, Auto-focusing and self-healing of Pearcey beams, Opt Exp, 20(2012)18955–18966.
- Chen Z G, Taflove A, Backman V, Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique, Opt Exp, 12(2004)1214–1220.
- Sheppard CJR, Intermediate field behind a nanostructure, Phys Rev A, 88(2013)033839; doi.org/10.1103/PhysRevA.88.033839.
- Zhu M, Cao Q, Gao H, Creation of a 50,000 λ long needle-like field with 0.36 λ width, J Opt Soc Am A, 31(2014)500–504.
- Sheppard CJR, Creation of a 50,000λlong needle-like field with 0.36λ width: comment, J Opt Soc Am A, 31(2014)1326–1327.
- Toraldo di Francia G, Supergain antennas and optical resolving power, Nuovo Cimento Supplemento, 9(1952)426–435.
- Toraldo di Francia G, Nuovo pupille superrisolvente, Atti Fond Giorgio Ronchi, 7(1952)366–372.
- Chen Z G, Taflove A, Backman V, Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique, Opt Exp, 12(2004)1214–1220.
- Sheppard C J R, Intermediate field behind a nanostructure, Phys Rev A, 38(2013)033839; doi.org/10.1103/PhysRevA.88.033839.
- Zhu M, Cao Q, Gao H, Creation of a 50,000λ long needle-like field with 0.36λ width, J Opt Soc Am A, 31(2014)500–504.
- Sheppard C J R, Creation of a 50,000λ long needle-like field with 0.36λ width: comment, J Opt Soc Am A, 31(2014)1326–1327.
- Toraldo di Francia G, Supergain antennas and optical resolving power, Nuovo Cimento Supplemento, 9(1952)426–435.
- Toraldo di Francia G, Nuovo pupille superrisolvente, Atti Fond Giorgio Ronchi, 7(1952)366–372.
- Sales, T R M, Smallest focal spot, Phys Rev Lett, 81(1998)3844–3847.
- Wu M, Chen R, Soh J, Shen Y, Jiao L, Wu J, Chen X, Ji R, Hong M, Super-focusing of center-covered engineered microsphere. Sci Rep, 6(2016)31637; doi.org/10.1038/srep31637.
- Wang Z, Guo W, Lim L, Luk’yanchuk B, Khan A, Liu Z, Chen Z, Hong M H, Optical virtual imaging at 50nm lateral resolution with a white-light nanoscope, Nature Commun, 2(2011)218; doi.org/10.1038/ncomms1211
