Evolution Law of Effective Radius of the Optical Beam
The effective (root-mean-square) radius of the beam Re (usually called instantaneous effective radius) is the second-order centered moment, and its square can be expresses as
where r± = exx + ey is the transverse coordinate; ex, ey are the directing orths; I = |U | 2cn0/8n is the radiation intensity; U is the strength of the electric field of wave; P = /r h d2r±I(r±, z) is the radiation power; n0 is the (linear) refractive index of the medium; is the region of determination of the transverse beam intensity profile
is the radius vector of the centroid. In contrast to the geometric radius of the beam measured at the given intensity level and reflecting the actual transverse scale of the beam only for the unimodal type of distribution, the effective radius determines the characteristic transverse size of the zone, where the most part of the radiation energy is concentrated for any beam intensity profile.
The solution of the equation of quasi-optics for the linear medium leads to the versatile evolution dependence of the effective radius of the focused optical beam at any point z of the optical path :
is the effective diffraction divergence of the collimated beam of the transverse intensity profile; F is the initial wavefront curvature radius; Re0 = Re(z = 0); P0 = P(z = 0); k0 = 2n/A0 is wave number; A0 is the radiation wavelength. It should be noted that the diffraction divergence 6D is connected with the so-called beam quality parameter M2 = 0Dk0Re0 = 0D/6Dg, demonstrating how many times the angular divergence of the studied beam is larger than the diffraction divergence of the Gaussian beam 6Dg = 1/k0Re0 of the same initial effective radius Re0.
At the stationary self-focusing of the laser beam in the medium with the cubic Kerr nonlinearity taking place at P0 > P, where Pc is the critical power, there also exists the average description of the process similar to Equation 4.74, which is valid at least up to the point of transverse collapse of the beam zN (nonlinear focus). In this case, for every particular type of the beam, it is sufficient to calculate few parameters (Re0,9D,P) which depend only on the optical parameters of the medium and the transverse profile of the electric field strength of the wave, to determine the stationary effective beam radius Re:
where n0 = P0/Pc is the self-focusing parameter, and critical self-focusing power is determined as
(n2 is the nonlinear addition to the refractive index of the medium due to the Kerr effect). The parameters given by Equations 4.73, 4.75, and 4.77 describes the coefficients of radiation propagation in the nonlinear medium, and they depend on the laser beam profile.
We can transform Equation 4.76 to the following form:
Here, the dimensionless variables z = z/LD, Re = Re /Re0, F = F/LD, are introduced, and the parameter LD = Re0/9D has the meaning of the effective diffraction length of the beam with the given spatial intensity profile. It follows from Equation 4.78, in particular, which at n0 > 1 the laser beam in general (in the sense of effective radius) can compress to a point at the distance:
where zK = LD Fjn<> — 1 is the coordinate of the point of transverse Kerr collapse for the collimated radiation. After the point of nonlinear focus zN, Equation 4.78 makes no sense.
The type of transverse intensity distribution of an optical beam was not ever specified yet. This means that, in relative coordinates with allowance for the corresponding change of the relative radiation power n0, Equation 4.78 does not lose its versatility for any type of beams. In the further discussions, we consider three most often practically used types of optical beams: Gaussian (GB), super-Gaussian (SGB), and ring beam (RB).
For the beam with the Gaussian initial profile, the electric field strength of optical wave is described by the function of the form
with the geometric radius R0 (at the 1/e level of intensity maximum). Then the initial values of the effective GB parameters are Re0 = R0; 6D = dDg = 1/(k0R0), Ld = Lds = ko Ro2 and Pc = Pcg = Ао2/(2ппоП2 )
The beam of the super-Gaussian profile at the circular aperture
is characterized by different values of the critical power, diffraction length, and divergence. And, finally, the third type of the beam intensity distribution—the ring beam—is described by the difference of two Gaussians
where p = RJR0 is the shadowing parameter; Rh is the radius of the shadowed part of the beam at the e-1 intensity level.
Numerical estimates of the propagation coefficients for beams of different transverse profiles (4.80 through 4.82) have shown that the critical power and the quality parameter take minimal values of the Gaussian beam (G = M2 = 1 at q = 2, p = 0), whereas the initial effective radius of SGB is always smaller than the beam radius of the Gaussian beam. Ring beams have the largest initial effective radius.