Abstract

Since the development of laser light sources in the early 1960s, laser beams are everywhere. Laser beams are central in many industrial applications and are essential in ample scientific research fields. Prime scientific examples are optical trapping of ultracold atoms, optical levitation of particles, and laser-based detection of gravitational waves. Mathematically, laser beams are well described by Gaussian beam expressions. Rather well covered in the literature to date are basic expressions for scalar Gaussian beams. In the past, however, higher accuracy mathematics of scalar Gaussian beams and certainly high-accuracy mathematics of vectorial Gaussian beams were far less studied. The objective of the present review then is to summarize and advance the mathematics of vectorial Gaussian beams. When a weakly diverging Gaussian beam, approximated as a linearly polarized two-component plane wave, say (Ex,By), is tightly focused by a high-numerical-aperture lens, the wave is “depolarized.” Namely, the prelens (practically) missing electric field Ey,Ez components suddenly appear. This is similar for the prelens missing Bx,Bz components. In fact, for any divergence angle (θd<1), the ratio of maximum electric field amplitudes of a Gaussian beam Ex:Ez:Ey is roughly 1:θd2:θd4. It follows that if a research case involves a tightly focused laser beam, then the case analysis calls for the mathematics of vectorial Gaussian beams. Gaussian-beam-like distributions of the six electric–magnetic vector field components that nearly exactly satisfy Maxwell’s equations are presented. We show that the near-field distributions with and without evanescent waves are markedly different from each other. The here-presented nearly exact six electric–magnetic Gaussian-beam-like field components are symmetric, meaning that the cross-sectional amplitude distribution of Ex(x,y) at any distance (z) is similar to the By(x,y) distribution, Ey(x,y) is similar to Bx(x,y), and a 90° rotated Ez(x,y) is similar to Bz(x,y). Components’ symmetry was achieved by executing the steps of an outlined symmetrization procedure. Regardless of how tightly a Gaussian beam is focused, its divergence angle is limited. We show that the full-cone angle to full width at half-maximum intensity of the dominant vector field component does not exceed 60°. The highest accuracy field distributions to date of the less familiar higher-order Hermite–Gaussian vector components are also presented. Hermite–Gaussian E-B vectors only approximately satisfy Maxwell’s equations. We have defined a Maxwell’s-residual power measure to quantify the approximation quality of different vector sets, and each set approximately (or exactly) satisfies Maxwell’s equations. Several vectorial “applications,” i.e., research fields that involve vector laser beams, are briefly discussed. The mathematics of vectorial Gaussian beams is particularly applicable to the analysis of the physical systems associated with such applications. Two user-friendly “Mathematica” programs, one for computing six high-accuracy vector components of a Hermite–Gaussian beam, and the other for computing the six practically Maxwell’s-equations-satisfying components of a focused laser beam, supplement this review.

© 2019 Optical Society of America

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2018 (3)

Y. Zhou, J. Zhao, Z. Shi, S. M. H. Rafsanjani, M. Mirhosseini, Z. Zhu, A. E. Willner, and R. W. Boyd, “Hermite–Gaussian mode sorter,” Opt. Lett. 43, 5263–5266 (2018).
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R. Zang, B.-Z. Wang, S. Ding, and Z.-S. Gong, “Evanescent-wave reconstruction in time reversal system,” Frequenz 72, 285–292 (2018).
[Crossref]

C. Bradac, “Nanoscale optical trapping: a review,” Adv. Opt. Mater. 6, 1800005 (2018).
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2017 (2)

2016 (6)

C. Bond, D. Brown, A. Freise, and K. A. Strain, “Interferometer techniques for gravitational-wave detection,” Living Rev. Relativity 19, 1–217 (2016).
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P. W. Milonni, “Lasers: the first 50 years. Scope: review, general interest. Level: general readership, teacher, undergraduate,” Contemp. Phys. 57, 113–116 (2016).
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U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Prog. Opt. 61, 237–281 (2016).
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U. Levy and Y. Silberberg, “Weakly diverging to tightly focused Gaussian beams: a single set of analytic expressions,” J. Opt. Soc. Am. A 33, 1999–2009 (2016).
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U. Levy and Y. Silberberg, “Weakly diverging to tightly focused Gaussian beams: a single set of analytic expressions,” J. Opt. Soc. Am. A. 33, 1999–2009 (2016) [Supplementary Material].
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S. M. Barnett, L. Allen, R. P. Cameron, C. R. Gilson, M. J. Padgett, F. C. Speirits, and A. M. Yao, “On the natures of the spin and orbital parts of optical angular momentum,” J. Opt. 18, 064004 (2016).
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2015 (7)

B. Major, Z. L. Horváth, and M. A. Porras, “Phase and group velocity of focused, pulsed Gaussian beams in the presence and absence of primary aberrations,” J. Opt. 17, 065612 (2015).
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J. Dressel, K. Y. Bliokh, and F. Nori, “Spacetime algebra as a powerful tool for electromagnetism,” Phys. Rep. 589, 1–71 (2015).
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U. Levy and Y. Silberberg, “Second and third harmonic waves excited by focused Gaussian beams,” Opt. Express 23, 27795–27805 (2015).
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S. V. Ershkov, “Exact solution of Helmholtz equation for the case of non-paraxial Gaussian beams,” J. King Saud Univ. Sci. 27, 198–203 (2015).
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C. L. Mueller, P. Fulda, R. X. Adhikari, K. Arai, A. F. Brooks, R. Chakraborty, V. V. Frolov, P. Fritschel, E. J. King, and D. B. Tanner, “In situ characterization of the thermal state of resonant optical interferometers via tracking of their higher-order mode resonances,” Classical Quantum Gravity 32, 135018 (2015).
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S. Soaidin, N. Arsad, R. M. Yeh, M. S. Ab Rahman, and S. Shaari, &quot;Analysis of multi-mode erbium doped fiber laser system,&quot; J. Opt. Adv. Mater. 17, 1278–1282 (2015).

U. Levy and Y. Silberberg, “Free-space nonperpendicular electric–magnetic fields,” J. Opt. Soc. Am. 32, 647–653 (2015).
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2014 (1)

J. C. Rautio, “The long road to Maxwell’s equations,” IEEE Spectrum 51, 36–56 (2014).
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2013 (3)

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys. 15, 033026 (2013).
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I. Bialynicki-Birula and Z. Bialynicka-Birula, “The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism,” J. Phys. A 46, 053001 (2013).
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L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre-Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
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2012 (2)

2011 (3)

F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express 19, 25143–25150 (2011).
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S. M. Barnett, “On the six components of optical angular momentum,” J. Opt. 13, 064010 (2011).
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J. Guerrero, F. F. López-Ruiz, V. Aldaya, and F. Cossío, “Harmonic states for the free particle,” J. Phys. A 44, 445307 (2011).
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2010 (6)

H. Kang, B. Jia, and M. Gu, “Polarization characterization in the focal volume of high numerical aperture objectives,” Opt. Express 18, 10813–10821 (2010).
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A. Freise and A. S. Kenneth, “Interferometer techniques for gravitational-wave detection,” Living Rev. Relativity 13, 1–81 (2010).
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B. M. Rodríguez-Lara, “Normalization of optical Weber waves and Weber-Gauss beams,” J. Opt. Soc. Am. A 27, 327–332 (2010).
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S. S. Bulanov, T. Z. Esirkepov, A. G. Thomas, J. K. Koga, and S. V. Bulanov, “Schwinger limit attainability with extreme power lasers,” Phys. Rev. Lett. 105, 220407 (2010).
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J. Hecht, “A short history of laser development,” Appl. Opt. 49, F99–F122 (2010).
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M. Rose, “A history of the laser: a trip through the light fantastic-presenting a timeline of notable laser-related scientific accomplishments,” Photon. Spectra 44, 58 (2010).

2009 (4)

2008 (2)

2007 (6)

E. Y. S. Yew and C. J. R. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275, 453–457 (2007).
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B. J. Smith and M. G. Raymer, “Photon wave functions, wave-packet quantization of light, and coherence theory,” New J. Phys. 9, 414–441 (2007).
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Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B 86, 319–326 (2007).
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K. Kincade, “Raydiance takes a new approach to ultrafast lasers,” Laser Focus World 43, 67–68 (2007).

H. Luo, S. Liu, Z. Lin, and C. T. Chan, “Method for accurate description of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 32, 1692–1694 (2007).
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K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53–56 (2007).
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2006 (11)

A. C. T. Wu and C. N. Yang, “Evolution of the concept of the vector potential in the description of fundamental interactions,” Int. J. Mod. Phys. A 21, 3235–3277 (2006).
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H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express 14, 2095–2100 (2006).
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K. Volke-Sepulveda and L. K. Eugenio, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A 8, 867–877 (2006).
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R. I. Hernandez-Aranda, J. C. Gutiérrez-Vega, M. Guizar-Sicairos, and M. A. Bandres, “Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems,” Opt. Express 14, 8974–8988 (2006).
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Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 31, 2619–2621 (2006).
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Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” New J. Phys. 8, 133–149 (2006).
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G. F. Calvo, A. Picón, and E. Bagan, “Quantum field theory of photons with orbital angular momentum,” Phys. Rev. A 73, 013805 (2006).
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B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
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E. Y. Yew and C. J. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express 14, 1167–1174 (2006).
[Crossref]

A. R. Zakharian, P. Polynkin, M. Mansuripur, and J. V. Moloney, “Single-beam trapping of micro-beads in polarized light: numerical simulations,” Opt. Express 14, 3660–3676 (2006).
[Crossref]

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Beams of electromagnetic radiation carrying angular momentum: the Riemann–Silberstein vector and the classical–quantum correspondence,” Opt. Commun. 264, 342–351 (2006).
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2005 (3)

O. Keller, “On the theory of spatial localization of photons,” Phys. Rep. 411, 1–232 (2005).
[Crossref]

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005).
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O. Steuernagel, “Equivalence between focused paraxial beams and the quantum harmonic oscillator,” Am. J. Phys. 73, 625–629 (2005).
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2004 (2)

2003 (2)

A. Brillet, J.-Y. Vinet, V. Loriette, J.-M. Mackowski, L. Pinard, and A. Remillieux, “Virtual gravitational wave interferometers with actual mirrors,” Phys. Rev. D 67, 102006 (2003).
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I. Bialynicki-Birula and Z. Bialynicka-Birula, “Vortex lines of the electromagnetic field,” Phys. Rev. A 67, 062114 (2003).
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2002 (4)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
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U. Levy and K. Wang, “High-order-mode fiber-based dispersion management technology yields full-slope match, high power tolerance, and low loss,” Proc. SPIE 4906, 280–285 (2002).
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Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
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Y. I. Salamin, G. R. Mocken, and C. H. Keitel, “Electron scattering and acceleration by a tightly focused laser beam,” Phys. Rev. Spec. Top. Accel. Beams 5, 101301 (2002).
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2001 (4)

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B 4, S7–S16 (2001).
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M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8, 458–474 (2001).
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N. N. Rosanov, V. E. Semenov, and N. V. Vyssotina, “‘Optical needles’ in media with saturating self-focusing nonlinearities,” J. Opt. B 3, S96–S99 (2001).
[Crossref]

N. Huse, A. Schoenle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 273–276 (2001).
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2000 (8)

B. M. Kim, J. Eichler, K. M. Reiser, A. M. Rubenchik, and L. B. Da Silva, “Collagen structure and nonlinear susceptibility: effects of heat, glycation, and enzymatic cleavage on second harmonic signal intensity,” Lasers Surg. Med. 27, 329–335 (2000).

R. Carminati, J. J. Saenz, J.-J. Greffet, and M. Nieto-Vesperinas, “Reciprocity, unitarity, and time-reversal symmetry of the S matrix of fields containing evanescent components,” Phys. Rev. A 62, 012712 (2000).
[Crossref]

M. Padgett and L. Allen, “Light with a twist in its tail,” Contemp. Phys. 41, 275–285 (2000).
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K. Bahlmann and S. W. Hell, “Depolarization by high aperture focusing,” Appl. Phys. Lett. 77, 612–614 (2000).
[Crossref]

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. II. Numerical results,” J. Opt. Soc. Am. A 17, 2090–2095 (2000).
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U. Levy and Y. Danziger, “High-order modes in high-capacity optical networks,” Proc. SPIE 4087, 405–410 (2000).
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A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6, 1389–1399 (2000).
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A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
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1999 (2)

C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
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H.-C. Kim and Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
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1998 (1)

P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118(1998).
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1997 (3)

X. Zeng, C. Liang, and Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 2042–2047 (1997).
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A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
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M. A. M. Marte and S. Stenholm, “Paraxial light and atom optics: the optical Schrödinger equation and beyond,” Phys. Rev. A 56, 2940–2953 (1997).
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1996 (4)

P. Varga and P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett 21, 1523–1525 (1996).
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M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
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C. S. Milsted and C. D. Cantrell, “Vector effects in self-focusing,” Phys. Rev. A 53, 3536 (1996).
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I. Bialynicki-Birula, “Photon wave function,” Prog. Opt. 36, 245–294 (1996).
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1994 (5)

M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
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S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
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W. Winkler, R. Schilling, K. Danzmann, J. Mizuno, A. Rüdiger, and K. A. Strain, “Light scattering described in the mode picture,” Appl. Opt. 33, 7547–7550 (1994).
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W. L. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
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K. Svoboda and S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
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1993 (2)

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1987 (3)

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1981 (2)

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1979 (5)

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1978 (2)

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1961 (3)

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1909 (1)

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O. Svelto and D. C. Hanna, Principles of Lasers (Plenum, 1998), Vol. 4.

U. Levy, Y. Silberberg, and N. Davidson, “Resonator,” figshare, 2019, https://doi.org/10.6084/m9.figshare.9730235 .

KEDMI Scientific Computing, “Numerit: a high-level intuitive programming environment,” http://www.numerit.com/ .

U. Levy, Y. Silberberg, and N. Davidson, “EEE & BBB HGnm,” figshare, 2019, https://doi.org/10.6084/m9.figshare.9730229 .

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Supplementary Material (3)

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» Code 1       A computer code runs on “NUMERIT” intuitive computational environment.
» Code 2       A “Mathematica” computer code.
» Code 3       A “Mathematica” computer code.

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Figures (36)

Figure 1.
Figure 1. Vector Gaussian beam flow chart. Starting with a scalar Gaussian beam, exactly or approximately satisfying the Helmholtz equation, vector Gaussian beams can be derived in several different ways. One way is to identify two of the six electromagnetic components with Gaussian beams and, guided by Maxwell’s equations, correctly calculate the other four components. Another way is to derive the electromagnetic field components through a Gaussian-seeded vector potential.
Figure 2.
Figure 2. Gaussian beam parameters. Propagation of the scalar fundamental Gaussian beam is uniquely determined by only two parameters [96], for example, the wavenumber in the medium (kH) and the waist (w0)
Figure 3.
Figure 3. Fundamental scalar GSN (absolute value of, not absolute value squared). A, on the xy plane at z=0. B, on the xz plane at y=0. {λH=1, w0=4·λH}. All length units are micrometers that are practically wavelength units.
Figure 4.
Figure 4. LP (“linearly polarized”) fiber mode [116], and an LG free space mode. A, very high-order LP mode at the end face of a multimode optical fiber, experimentally excited with a He–Ne laser (633 nm) [117,118]. B, mathematical LG4,26. The two modes are similar but NOT identical. Noticeably different is the outer ring (as expected).
Figure 5.
Figure 5. Gouy phase slowing of overall phase accumulation during Hermite-GSN propagation. Upon propagation across the entire z span, the accumulated phase of the beam is reduced by (N+1)·π (the phase at plane z is ϕ(z)=kH·z(N+1)·ψ(zN)) [111]. The parameter N is the combined mode number. Note that if the fast phase accumulation of the Hermite-GSN is described by exp(i·kH·z) [to go with exp(+i·ω·t)], then the curves in the figure change sign (i.e., are flipped around the horizontal zero line as in Chap. 10 of [105]).
Figure 6.
Figure 6. Scalar Hermite-GSN (3,0) (including the fast phase term ψ(x,y,z)=HGmn(x,y,z)·ei·kH·z). A, real part on the x, y plane at z=0 (imaginary part is zero). B, on the xz plane–absolute value of absolute value of the mode. C, on the xz plane–the real part, near the focal plane.
Figure 7.
Figure 7. Approximation accuracy of the high-order symmetric GSN analytic expressions (cf. Table 1) versus beam waist (w0/λH). The curves show Maxwell’s-residual powers, normalized by the power of the dominant electric field component (P(|Ex|2)). The low-valued curves indicate high accuracy of the approximation (to the exact GSN vectors) down to wavelength-scale spot sizes.
Figure 8.
Figure 8. LP asymmetric Gaussian field components at the focal plane. The field components were derived through a Lorenz-gauged vector potential [Eq. (19)] [40,41]. Ratio of maximum amplitudes is Ex:Ez:Ey1,301,9001 {w0=4·λH}.
Figure 9.
Figure 9. LP symmetric Gaussian field components at the focal plane. The field components were derived through a Lorenz-gauged magnetic vector potential [Eq. (19)] and were subsequently symmetrized [67]. Several published papers indeed support symmetrical field components (nonzero Bx; cf. Subsection 5.4). For the symmetric description, the ratio of maximum amplitudes is Ex:Ez:Ey1,301,18001 {w0=4·λH}.
Figure 10.
Figure 10. Fundamental LP GSN electric fields away from the focal plane (z=4·zR). Top row, real part of the fields. Bottom row, normalized real part (by the norm of the component). The force in each direction exerted on a stationary point electric charge, at any given time, periodically oscillates (including direction reversal) with distance from the symmetry axis. {w0=4·λH}.
Figure 11.
Figure 11. Angles between the electric–magnetic vectors of a LP asymmetric GSN. A, at the focal plane. B, at z=100. The radius (R) of the black circle in A is R=5·w0. Within the circle, the angles swing from 70° to 110°. The power outside the circle is 2·1022 of the total beam power. {w0=4·λH}.
Figure 12.
Figure 12. Angles between the electric field vector (E(r)) and the symmetry axis (z^). A, near the focal plane. B, further away. The radius (R) of the black circle in A is R=3·w0. Within the circle, the angles swing from 80° to 100°. The power outside the circle is 2·108 of the total beam power. {w0=4·λH}.
Figure 13.
Figure 13. Power conservation of the symmetric GSN electric–magnetic fields ([67] and Table 1 above; cf. the figure of Subsection 7.4 below showing the more general case of power conservation by Hermite-GSNs, and see also the relative power curves of Subsection 8.3). Power conservation assures accurate description of Gaussian beams at any axial distance. Several literature-published GSN distributions do not conserve power, and their accuracy is thus limited to the near-focal-plane region (about ±1 Rayleigh range).
Figure 14.
Figure 14. LP high-accuracy asymmetric Hermite–Gaussian (2,1) field components at the focal plane. The field components were derived through a Lorenz-gauged vector potential [Eq. (19)]. Note the additional zero lines in the distributions of the secondary components, much like the added zero lines for the secondary components of the fundamental vectorial GSN (cf. Fig. 8). {w0=4·λH}.
Figure 15.
Figure 15. LP high-accuracy symmetric Hermite–Gaussian (2,1) field components at the focal plane. The field components were derived through a Lorenz-gauged vector potential [Eq. (19)] and were subsequently symmetrized [Eq. (23)]. The ExBy and the EyBx symmetry stands out. Several published papers indeed support symmetrical field components (nonzero Bx). {w0=4·λH}. A user-friendly “Mathematica” computer program for calculation of high-accuracy symmetric HGmn field components supplements our review (cf. Appendix B).
Figure 16.
Figure 16. Angles between the electric–magnetic vectors of a LP asymmetric Hermite-GSN (2,1). A, real amplitude of Ex,sym at z=10. B, angles—large view. The radius (R) of the black circle in B is R=5·w0. Within the circle, except near the sign-crossing planes, the angles swing from 80° to 100°. The small (or large) angles at the edges of the view are between extremely small amplitudes of the vector fields. C, zoom-in to the sign-crossing planes. D, further zoon-in to a single sign-crossing plane. Note that whereas the angles between the electric–magnetic vectors are very small (or very large), the fields’ amplitudes are very small, approaching zero at the sign-crossing plane. {w0=4·λH}.
Figure 17.
Figure 17. Angles between the electric field vector (E(r)) and the symmetry axis (z^) for the asymmetric Hermite-GSN (2,1) (cf. Fig. 16(A) for real amplitude distribution). A, at z=10, narrow view. B, at z=10, wider view. The radius (R) of the black circle in B is R=3·w0. C, at y=2.8 in the xz plane. D, at z=100 on the xy plane. Away from the sign-crossing planes, axial angles of the electric field vector swing between θ80° and θ100°. {w0=4·λH}.
Figure 18.
Figure 18. Power conservation [Eq. (31)] of Hermite-GSN (2,1). A, asymmetric set (cf. Fig. 14). B, symmetric set (cf. Fig. 15). C, symmetric–analytic set (cf. Appendix B). Power conservation assures accurate description of Hermite-GSN beams at any axial distance.
Figure 19.
Figure 19. Hermite-GSN (2,1), distance evolution of the electric field intensity distributions. Top row, cross-sectional intensity shape of the dominant component. Distribution proportionally expands but otherwise stays fixed. Center and bottom rows, visible changes in cross-sectional intensity shapes of the secondary components with propagation distance [48]. {w0/λH=2;zR=12.5}.
Figure 20.
Figure 20. Hermite-GSN (2,1), distance evolution of magnetic field intensity distributions. Shown are three distance snapshots for the axial magnetic field component (Bz). For intensity evolutions of the other two components, see Fig. 19 with ExBy and EyBx{w0/λH=2;zR=12.5}.
Figure 21.
Figure 21. Hermite-GSN (2,1), distance evolution of intensity distributions. Shown are three distance snapshots for the seed function (top row) and for the dominant electric field component (bottom row). Mathematically, at very small waist (and thus poor paraxial approximation) and near the focal plane, cross-sectional intensity distributions of the dominant component also change their shapes with distance. {w0=0.5·λH;zR=0.78}.
Figure 22.
Figure 22. Focal plane distributions of the exact scalarGSN-like solution of Helmholtz equation [the “seed function,” Eq. (32)]. A, the homogeneous part (ψH(x,y,0)). B, the evanescent part (ψev(x,y,0)). At z=0, both distributions are real. {w0=1.35·λH}.
Figure 23.
Figure 23. LP asymmetric exact Gaussian-like field components at the focal plane. The field components were derived through a Lorenz-gauged vector potential [Eq. (19) with the magnetic vector potential of Eq. (33)]. These Gaussian-like electromagnetic fields rigorously satisfy Maxwell’s equations. At the selected relatively large waist, the exact distributions shown here are very similar (but not identical) to the approximate distributions shown in Fig. 8. As the seed waist (w0) shrinks, “visible” differences do arise (see below the maps for very small seed waist of w0=0.025·λH). {w0=4·λH}.
Figure 24.
Figure 24. LP symmetric practically exact Gaussian-like field components at the focal plane. The field components were derived through a Lorenz-gauged vector potential [Eq. (19) with the magnetic vector potential of Eq. (33)] followed by symmetrization [Eq. (23)]. For all practical purposes, these Gaussian-like symmetric electromagnetic fields satisfy Maxwell’s equations. At the sub-half-wavelength waist selected for the figure {w0=0.4·λH}, the Ex and the By components are elliptic [55,67,144]. A supplemental user-friendly program for calculating these E-B components is described in Appendix C.
Figure 25.
Figure 25. “Seed” exact scalar function [Eq. (32)] and the x component of the symmetric nearly exact Gaussian-like fields for an extremely small seed waist. Top row, the complete solution. Bottom row, homogeneous part only (evanescent part excluded). A and D, the seed scalar function at z=0. B and E, x component of the electric field at z=0. C and F, far-field distribution of the x component of the electric field. At the selected extremely small waist (for the seed function, w0=0.025·λH), the near-field distributions are markedly different (A versus D and B versus E). In contrast, the “far”-field distributions are indistinguishable (C versus F). In both far-field cases, evanescent waves do not contribute any more. Note that time reversal of the wave from any z>0 plane will not restore the original z=0 distribution. Experimental reconstruction of the evanescent waves to achieve “super resolution” requires special external intervention [178]. It follows that if the evanescent waves are included in the overall description of the electromagnetic waves, the electromagnetic waves no longer propagate in free space from minus infinity to plus infinity. The description is restricted to waves in the z>0 plane with a source at z=0.
Figure 26.
Figure 26. Focal plane spatial distributions of the vector components of the homogeneous (propagating) waves and of the evanescent waves of the exact (Maxwell’s-equations-solving) Gaussian-like beam. As with all other shown distributions, the beam is “LP” in the x direction. The waist (of the seed function) is taken to be rather small (w0=0.2·λH) to emphasize the contribution of the (strong) evanescent waves. A, homogeneous (propagating) waves. Top row, real; bottom row, imaginary (no absolute value row in this panel). As is the case with all approximate vectorial Gaussian beams, the x, y components of the propagating waves at the focal plane are purely real, whereas the z component is purely imaginary (cf. Fig. 9 for example, and consult Table 1). Note also the relative amplitude strengths. B, evanescent waves. Top row, real; bottom row, imaginary (no absolute value row in this panel). Here the y component is still purely real, but the other two components are mixed. Pay attention to the relative strengths of the amplitudes and compare with the strengths shown in A. C, the two wave-types together (the full distributions). Bottom row, absolute value of the full distributions as noted. The “smallness” of the distributions of the fields in the bottom row [relative to the “largeness” of the distributions in (A)] results from the interference of the propagating waves (A) with the strong evanescent waves (B). Pay attention also to the (barely visible) evanescent rings surrounding the four lobes of the y component shown in the middle of the bottom row of C. All shown 21 squares of the figure extend between 1.6 and 1.6 (essentially in wavelength units).
Figure 27.
Figure 27. Power conservation of the nearly exact symmetric vector GSN. A, total power transported along the propagation axis [integrated z component of the Poynting vector—Eq. (31)]. For the shown symmetric GSN, there is an invisible, insignificant dip of the straight line near z=0 (|pev(z)/phom|<3·107). B, the axial (z) component of the time-averaged Pointing vector [Eq. (30)], calculated for the exact, asymmetric evanescent waves. The calculated Poynting vector component, shown for a specific xy plane, is consistently zero. In other words, as is well known, energy is not transported “forward” by evanescent waves [176,177]. {w0=0.5·λH}.
Figure 28.
Figure 28. Relative powers carried by the vector components of the Gaussian beam versus waist-to-wavelength ratio. The solid curves are computed by the expressions of Eq. (34) for the fundamental vectorial GSN (m=n=0). The dots in the two panels were numerically calculated by us for the nearly exact symmetric Gaussian-like components discussed in this section. The relative powers do not depend on the axial distance (z) but only on the waist-to-wavelength ratio [Eq. (34)]. A, relative power carried by the longitudinal component. B, relative power carried by the weak transverse component (either PEy to PEX or PBx to PBy). As shown, the analytic paraxial expressions [Eq. (34)] well predict the relative power of the longitudinal component even for the nearly exact case, and similarly well predict the power carried by the nearly exact weak transverse component. At waist of half-wavelength, the power carried by the longitudinal component is 10% of the power carried by the strong transverse component, and only 0.3% is carried by the weak transverse component. The curves shown in this figure are important for studies where vector fields are involved, allowing researchers to know up-front rather accurately the relative weight of powers carried by the field components, given the waist-to-wavelength ratio.
Figure 29.
Figure 29. FWHM-intensity divergence angles for the symmetric nearly exact Gaussian field components versus normalized waist (w0/λH). The shown six curves are as follows: (1) divergence of a fundamental scalar GSN; (2) divergence of the seed Gaussian-like function; (3) divergence of the x component of the nearly exact electric field in the x direction; (4) divergence of the x component of the nearly exact electric field in the ydirection; (5) the analytic equation 360π·arctan{λH·[ln(2)]1/2π·w0}; (6) curve according to a 1997 analytic expression for the far-field distribution of a scalar Gaussian-like amplitude, rigorously solving HE3D, derived by Zeng et al. [96]. The horizontal dotted brown line is set at 65.5° according to the prediction by Zeng et al. [96]. All curves but 5 were generated by solving an implicit equation for the radial distance at half-intensity (and knowing the corresponding distance). A, full view. Looking at curves 1 and 5, the full-cone divergence of a theoretical (nonphysical) pure GSN goes to 180° as the waist goes to zero. Looking at curves 2 and 6, the full-cone divergence angle at FWHM intensity of an exact scalar GSN-like solution of the Helmholtz equation [Eq. (32)] is limited to 65.5° (NA of 0.54 when measured at FWHM intensity). Looking at curve 3, divergence angle of Ex(x) is even smaller, limited to about 51° (NA=0.43 if truncated at FWHM intensity). Looking at curve 4, since the source (Ex) is “horizontally elliptic,” the far-field divergence angle of Ex(y) is somewhat larger, limited to about 59° (NA=0.49 if truncated at FWHM intensity). B, zoom-in to very small waists. Note that for w0>λH, all curves practically merge.
Figure 30.
Figure 30. FWHM intensity divergence angles for the symmetric nearly exact and approximate Gaussian field components versus normalized waist (w0/λH). The shown six curves are as follows: (1) divergence of a fundamental GSN; (2) divergence of the seed Gaussian-like function; (3) divergence of the x component of the nearly exact electric field in the x direction; (4) divergence of the x component of the nearly exact electric field in the y direction; (5) divergence of the x component of the approximate symmetric electric field in the x direction [symmetrization—Eq. (23) with HG00, Eq. (24) as seed]; (6) divergence of the x component of the approximate symmetric electric field in the y direction [symmetrization—Eq. (23) with HG00, Eq. (24) as seed]. The horizontal dotted brown line is set at 65.5° according to the prediction by Zeng et al. [96]. For the first four curves, see Fig. 29. The added curves 5,6 depict the divergence of Ex,sym as derived through the Lorenz-gauged vector potential [Eq. (23) and cf. the odd-numbered lines of Table 1]. The seeded scalar function is the fundamental GSN [Eq. (24)] (the first term in a series expansion of the solution to Helmholtz equation by divergence angles [67]). As already stated above, at very tight focusing, the waist at the focal plane is elliptic. At the far-field, the spot is elliptic with 90° rotation (similar to those shown by Figs. 25(C) and 25(F)). The divergence angle curves “fall” on both sides of the fundamental GSN divergence curve, the x direction below, and the y direction above. Already at w0/λH=0.4, the approximate (analytic) GSN FWHM intensity cone angles are exaggerated relative to the nearly exact GSN FWHM cone angles by about +10° or +12°. For yet smaller spot sizes (of the GSN seed function), the approximation leading to the analytic solutions is no longer valid; therefore, the mathematically handy analytic expressions should not be called for.
Figure 31.
Figure 31. x component of the electric field, one of the six symmetric nearly exact Gaussian-like components, at the focal plane (z=0). A, field distribution with as-calculated evanescent component (invisible). B, visible evanescent “rings” after multiplication of the evanescent component by a factor of 2·104. The objective of the figure is to show that the near-field evanescent rings are always present, however weak, even at relatively large spot sizes (w0=λH in the figure). As the waist (of the seed function) shrinks, the relative weight of the evanescent rings shoots up (cf. Fig. 25(B)). Increased spot-blur with decreased waist, together with nongrowing divergence angle (cf. Fig. 29(B)), validate the classical Gaussian-beam-related uncertainty relation. The reader can test this behavior through manipulations of the Mathematica software code accompanying our review.
Figure 32.
Figure 32. Normalized profile of the fundamental-GSN, of the exact scalar GSN, and normalized profiles of the nearly exact Gaussian-like Ex,exact,sym versus the transverse coordinates. All curves are plotted for absolute amplitude distributions at a distance of z=40. The shown four curves are as follows: (1) profile of a fundamental GSN; (2) profile of the seeded Gaussian-like function [Eq. (32)]; (3) profile of the x component of the nearly exact electric field in the x direction; (4) profile of the x component of the nearly exact electric field in the y direction. The horizontal dotted black line is set at 1/2 (FWHM intensity). A and B, small/very small seed waist (as marked). At the shown small waists (selected for the seeded functions), far-field profiles 2,3,4 associated with the nearly exact Gaussian-like distributions are visibly different compared to the distribution of the paraxial fundamental scalar GSN—curve 1.
Figure 33.
Figure 33. Phase accumulation near the focal plane—scalar fundamental GSN (blue) and the nearly exact Gaussian-like x component of the electric field (red). The analytic Gouy phase (ψG(z)) is designated by the green line. A, relatively large waist (w0=2·λH). B, smaller waist (w0=λH). Close inspection of the crest spacings of the curves in B reveals slightly larger spacing near the focal plane [182] (versus the further-away spacings). This near-focal-plane wider spacing is of course the result of a lower phase accumulation rate (space wise) as predicted by Gouy [97,98]. In terms of phase velocity, near the focal plane, the phase velocity of GSNs slightly exceeds the phase velocity of a plane wave crossing the same medium [142]. If the GSN propagates in empty space, then its near-focal-plane phase velocity exceeds the speed of light.
Figure 34.
Figure 34. Early stage of evolution of a coherent optical wavefront, bouncing back and forth inside a confocal resonator. Wavefronts are calculated sequentially by a computer program supplementing our review. The user can select mirror diameter, light wavelength, mode number, and number of round trips. Following a random-phase start, the wavefront evolves through Fresnel–Kirchhoff diffraction integral until, after many round trips, the wavefront stabilizes at the selected resonator mode. Predetermined “weak” constraints introduced into the resonator assure convergence to a preselected mode.
Figure 35.
Figure 35. “Mathematica” code to calculate vectorial symmetric Hermite–Gaussian beams. The user will input the shown five parameters and run the program. The program calculates the six Hermite–Gaussian symmetric components based on high-accuracy analytic expressions. The program plots the calculated six distributions with the user’s selected resolution. Without plotting (optional), program execution by a common desktop computer ends in less than half of a second.
Figure 36.
Figure 36. “Mathematica” code to calculate vectorial nearly exact symmetric fundamental Gaussian-like beams. For all practical purposes, the two calculated Gaussian-like vectors (E-B) satisfy Maxwell’s equations. The user will input the shown three parameters and decide between four inclusion options—with/without homogeneous waves and with/without evanescent waves, and evaluate the notebook. The program calculates the six symmetric components based on the seed (32) derived in [2], and then follows the symmetrization procedure of Eq. (23). The program plots the calculated six distributions with the user’s selected resolution. Without plotting (optional), program execution by a common desktop computer ends in less than 3 s.

Tables (3)

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Table 1. High-Order Expressions (in the Divergence Angle) for the Symmetric Field Components of a Linearly Polarized Gaussian Beam [67]a

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Table 2. First-Order Expressions for the Symmetric Field Components of a Linearly Polarized Gaussian Beam [67]a

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Table 3. Approximation Accuracy of Symmetric GSN Analytic Expressions (Tables 1 and 2)a

Equations (39)

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E(r)=0,
B(r)=0,
×E(r)=i·k0·B(r),
×B(r)=i·k0·ϵ(1)·E(r).
2E(r)+kH2·E(r)=0;2B(r)+kH2·B(r)=0.
2ψ(r)+kH2·ψ(r)=0.
ψ(qx,qy,z)=U(qx,qy)·e±i·kz·z,t2[U(qx,qy)]+kt2·U(qx,qy)=0,kz(kt)=[kH2kt2]12.
ψ(x,y,z)F(x,y,z)·ei·kH·z,t2[F(x,y,z)]+i·2·kH·F(x,y,z)z=0.
PGSN(ρ,z)P0=[1e2·ρ2/w2(z)]power transmission
F1(x,y,z)G(ρ,z)=A1·w0w(z)·exp(ρ2w2(z))·exp(i·kH·ρ22·R(z))·exp(i·ψG(z)),
F2(x,y,z)G(ρ,z)=A2·1q(z)·exp(i·kH·ρ22·q(z)),
F3(x,y,z)G(ρ,z)=A3·μ(z)·exp(μ(z)·ρ2w02).
I(ρ,z)=I0·(w0w(z))2·exp(2·ρ2w(z)2);I(0,z)=I01+zN2;P=(12)·I0·(π·w02).
F4(x,y,z)HGmn(x,y,z)=A4·(w0w(z))·Hm(xσ(z))·Hn(yσ(z))·exp(ρ2w2(z))·exp(i·kH·ρ22·R(z))·exp[i·ψG,mn(z)],
um(x,z)=(2π)14(12m·m!·w0)12·(q0q(z))12·(q*(z)q(z))m2·Hm(2·xw(z))·exp(i·kH·x22·q(z)),F5(x,y,z)HGmn(x,y,z)=A5·um(x,z)·un(y,z).
(q0q(z))12·(q*(z)q(z))m2=(w0w(z))12·exp[i·(m+0.5)·ψG(z)].
B(r,t)=×A(r,t),E(r,t)=Φ(r,t)1c·A(r,t)t.
A(r,t)+ϵ(1)(ω)c·Φ(r,t)t=0.
E(r)=i·k0·{[AH(r)kH2(ω)]+AH(r)},B(r)=×AH(r).
2AH,j(r)+kH2·AH,j(r)=0;j=x,y,z.
E(r)0,×B(r)i·k0·ϵ(1)·E(r).
E(r2,ψ,z2)[I0(r2,z2)+I2(r2,z2)·cos(2·ψ)]·i+I2(r2,z2)·sin(2·ψ)·j+2·i·I1(r2,z2)·cos(ψ)·k,B(r2,ψ,z2)I2(r2,z2)·sin(2·ψ)·i+[I0(r2,z2)I2(r2,z2)·cos(2·ψ)]·j+2·i·I1(r2,z2)·sin(ψ)·k,
AH1(r)=ψseed(r)·[1,0,0];AH2(r)=ψseed(r)·[0,1,0];r(x,y,z),E1(r)=i·k0·{[AH(r)kH2(ω)]+AH1(r)};B1(r)=×AH1(r),E2(r)=k0kH·×AH2(r);B2(r)=i·kH·{[AH2(r)kH2(ω)]+AH2(r)},Esym(r)=12[E1(r)+E2(r)];Bsym(r)=12[B1(r)+B2(r)].
ψ00(x,y,z)=G(ρ,z)·ei·kH·z,G(ρ,z)=E0i·k0·μ(z)·exp(μ(z)·ρ2w02),AHLNR(x,y,z)=ψ00(x,y,z)·[1,0,0],
Vj={M1,M2,(M3)x,(M3)y,(M3)z,(M4)x,(M4)y,(M4)z};jF,H,
Pj(w0)=|Vj(x,y,z;w0)|2·dx·dy;jF,H.
PN=PP(|Ex(r)|2)=P|Ex(x,y,z)|2·dx·dy.
RH/F(w0)=PHN(w0)PFN(w0).
θ[deg.]=[180π]·cos1{Re[E(x,y,z)B*(x,y,z)]|E(x,y,z)|·|B(x,y,z)|}.
Savg(x,y,z)Re[E(x,y,z,t)×B*(x,y,z)].
P(z)=Savg,z(x,y,z)·dx·dy.
ψH(x,y,z)=limδ001δ1f2·eb22·f2·ei·kH·z·(1b2)12·J0(kH·ρ·b)·b·db,ψev(x,y,z)=limδ01+δ1f2·eb22·f2·ekH·z·(b21)12·J0(kH·ρ·b)·b·db,ψGSN-exact(x,y,z)=ψH(x,y,z)+ψev(x,y,z);f2kH·w0,
AH(x,y,z)=ψGSN-exact(x,y,z)·[1,0,0].
REzx(paraxial Gaussian-Maxwell)PzPx=(2·m+1)(2·π·w0λH)2,REyx(paraxial Gaussian-Maxwell)PyPx=(2·m+1)·(2·n+1)4·(2·π·w0λH)4.
KλHπ·1w86.5·θ86.51.
2A(r)ϵ(1)(ω)ϵ(1)(2·ω)·[kH(2·ω)]2·A(r)=i·8·π·k0·P(2)(r).
Px(2)=χQ·[x(ExEx(1/3)·E2)+y(ExEy)+z(ExEz)],Py(2)=χQ·[x(EyEx)+y(EyEy(1/3)·E2)+z(EyEz)],Pz(2)=χQ·[x(EzEx)+y(EzEy)+z(EzEz(1/3)·E2)].
[Px(2ω)Py(2ω)Pz(2ω)]=[00d3100d3200d330d240d1500000]=[Ex·ExEy·EyEz·Ez2·Ey·Ez2·Ex·Ez2·Ex·Ey].
IA=(Px(2ω))2·cos2α+(Py(2ω))2·sin2α+(Pz(2ω))2=(Ex·Ez)2·cos2α+(Ey·Ez)2·sin2α+(Ex2+Ey2+Ez2).

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