Elegant Laguerre–Gaussian beams—formulation of exact vector solution

Abstract

In photonic applications of optical beams, their transverse cross-section should be often narrow, with a diameter in their waist of the order of one wavelength or even less. Within this range, the paraxial approximation of beam fields is not valid and standard corrections by field expansions with respect to a small parameter are not efficient as well. Thus, still there is a need for more accurate beam field description. In this report, an exact vector solution for free-space propagation is given in terms of elegant Laguerre–Gaussian beams. The analysis starts from the known paraxial field approximation and next, through bidirectional field transformation and application of a Hertz potential leads to an exact vector solution. The role of the paraxial solution in construction of the exact solution is elucidated. The method works well not only in cases of free-space propagation but also in description of beam interactions with planar interfaces and multilayers.

A concise and universal method for deriving arbitrary paraxial and d’Alembertian cylindrical Gaussian-type light modes

“…(36) which accounts for the modified Bessel-Gaussian beam of the nth order, the profile of which contains the hyperbolic Bessel function. After applying (26) together with χ = z R sin θ the beam acquires its conventional form: [34]:…”

“…It is a rapidly expanding area of science due to its numerous and significant practical applications ranging from optical trapping and guiding through image processing, optical communication, quantum cryptography to biology and medicine [1][2][3][4][5][6][7][8]. One can mention here pure Gaussian beams [9][10][11][12][13][14][15][16][17][18] with and without vortex component, Bessel-Gaussian (BG) [13,[19][20][21] and Laguerre-Gaussian (LG) beams [10,13,[21][22][23][24][25][26] of various orders or Kummer-Gaussian (KG) (i.e., Hypergeometric-Gaussian) beams [27]. Viewed from a purely theoretical perspective, these mathematical expressions describing the aforementioned beams were derived directly by solving wave equations [9,14,15,17,18,28,29], using the Fresnel diffraction formula [19,30,31], via superposing Gaussian beams, Bessel beams or even plane waves [15,[32][33]…”

Constructing various paraxial beams out of regular and modified Bessel-Gaussian modes

“…Although the Gaussian beam (i.e the so called fundamental mode) in many situations accurately describes the laser field that exhibits cylindrical symmetry, many practical applications require somewhat more complex patterns. Enough to mention here other cylindrical beams as regular Bessel-Gaussian (BG) beams [11][12][13][14][15] or modified ones (mBG) [16] and Laguerre-Gaussian (LG) beams [3,11,15,[17][18][19][20][21] of various orders which exhibit ring-like structure, Kummer-Gaussian (KG) (i.e., Hypergeometric-Gaussian) beams [22,23] or non-cylindrical beams like for instance Hermite-Gaussian ones [1,3] or more general paraxial beam [24]. Due to their numerous applications ranging from pure physics through optical communication technologies, image processing, up to biology and medicine [25][26][27][28][29][30][31][32][33][34], the structured light has earned an extensive literature cited above only in a very nutshell.…”

Paraxial Dirac equation

“…Alternatively one can say that the momenta perpendicular to the main axis are negligible when compared with the longitudinal component. The resulting paraxial equation, which in the scalar form is analogous to (3.5), has been widely explored providing rigorous solutions in the form of various beams: Gaussian beams [1,[3][4][5][6][7][8][9][10][11][12][13], regular Bessel-Gaussian (BG) beams [3,[14][15][16][17] and modified ones (mBG) [18] as well as Laguerre-Gaussian (LG) beams [3,4,17,[19][20][21][22][23] or Kummer-Gaussian (KG) (i.e. Hypergeometric-Gaussian) beams [24,25].…”