The measurement of the divergence (the width of the angular distribution) of the laser radiation serves to estimate the thermal load experienced by the optical elements of the laser. Measurement of the divergence (the width of the angular distribution) of radiation by the focal spot method, i. e. according to the size of the beam diameter, determined in accordance with the standard  in the focal plane of the collecting lens, gives an idea only of the total divergence. However, two main components contribute to divergence, in general: • physical θph, due to diffraction, the presence of many modes and inhomogeneities in the active medium; • geometric θg, caused by the curvature of the wave front at the output aperture of the laser. • The total divergence θt is the convolution of both components (both angular distributions), i. e. it is determined by the sum with respect to the quadratic law: θt2 = θph2 + θg2. The beam quality (its focus) is determined only by the physical divergence. The geometric divergence does not affect the quality of the beam, but in a number of applications of lasers (e. g., in optical locations) it must be carefully controlled in order to achieve the required total divergence.
For a separate measurement of the components of the divergence, it is necessary to determine the hyperbolic dependence of the beam diameter in the waist region formed when it is focused by a lens or a mirror (also as in determining the quality of the beam ). Figure 1 shows two variants of beam focusing with a plane and convex wave front at the output aperture of the laser. The same notation is given for the quantities required for the further presentation. In the absence of geometric divergence (for a flat front at the laser output) and for a lens not too far from the lens L, the plane wave front arrives at the lens and the waist plane practically coincides with the focal plane of the lens. The total divergence is physical and is determined by dividing the diameter dφ by the focal length of the lens F. In the presence of geometric divergence, a convex (in this example) wave front arrives at the lens and the plane of the waist is removed from the lens to the position Z0. Then the total divergence θt is determined by dividing the diameter dt by the focal length of the lens F (θt = dt/F), and the physical divergence θph is determined, in a first approximation, by dividing the waist diameter dph* by its removal Z0 from the lens (θph = dph* / Z0). The geometric divergence θg, in the first approximation, is determined by the formula: . It is advisable to refine the values of θph* and θg because they are obtained without taking into account the increase in the diameter of the beam incident on the lens in comparison with the diameter dA at the output aperture of the laser. This increase, especially at large values of θg and L, can lead to a significant underestimation of θph and, accordingly, to overestimation of θg. The refinement is achieved by multiplying θph by the correction factor (dA + θg · L) / dA, followed by recalculation of θg. If the corrections are large (more than 20% of the original values), the refinement of the newly obtained values is repeated once more. With the geometric divergence due to the convergent wave front at the output aperture of the laser, the waist plane is closer to the lens than the focal plane. The determination of the divergence θ and its components θph and θg is carried out in a similar way, the only difference being that the correction coefficient has the form (dA – θg · L) / dA. It should be noted that the Gaussian beam, even having a flat front at the output aperture of the laser, by self-diffraction, after passing the distance L to the lens, acquires the curvature of the wave front with a radius: , where λ is the wavelength of the laser radiation.
However, the numerical estimate for a real variant of the fiber laser parameters measuring circuit (for example, λ = 1.07 µm, dA = 12 mm, L = 10 m) gives a value of R more than 1 km, which makes a negligible difference between F and Z0. The above determination of the beam diameter dependence in the waist region is usually made by longitudinal scanning (successive measurement of the profile) of the beam along its axis using the profile analyzer. However, in high-energy lasers, due to thermal deformations in the optical scheme, the shape of the waist does not remain constant during the start-up. Therefore, its longitudinal scanning makes spatial parameters, defined in the form of a waist, not entirely reliable. Figure 2 shows qualitatively how the hyperbolic dependence of the diameter of the beam in the waist region varies with the physical and geometric divergence changing separately. The initial dependence is shown in black. The presence of significant initial geometric divergence (the front at the output aperture of the laser in this case is convex) manifests itself at a considerable distance from the focal plane of the lens F. As the physical divergence increases (for example, due to the increase in thermal inhomogeneities in the active medium), the diameter of the waist and hyperbola in the waist region (green) increases vertically upward. When the geometric divergence is changed (for example, due to the appearance of a thermal lens on the output optics of the laser), the waist Z0 of the hyperbola is displaced horizontally. With a positive thermal lens, Z0* approaches the focal plane F (red color), and the focal length of the thermal lens is determined by the formula: , where: Z0 is the position of the waist at the start of the start; Z0* is the position of the waist at the end of the start; L = –distance from the output aperture of the laser to the collecting lens.
With a negative thermal lens, Z0* is removed from the focal plane F (blue) and the focal length of the thermal lens is determined by the formula: . According to the figure, it is obvious that if only the total divergence (by diameter of beam d in the focal plane F) is controlled, it is impossible to establish the cause of its change, and take appropriate measures for its correction. The SIEPH device (a means of measuring the energy and spatial parameters of laser radiation) is used to measure the energy of pulsed laser radiation and the beam width of laser radiation. The principle of operation of the device is based on the conversion of the intensity distribution of laser radiation in the beam cross section into digital codes by a photosensitive matrix converter and their subsequent processing by a computer. The SIEPCH‑2 meter, equipped with two measuring chambers, allows controlling the entire hyperbolic dependence of the beam diameter in the waist region in time due to the simultaneous measurement of the profile in four planes in this region. A variant of the corresponding scheme using four optical wedges, two diffuse reflecting screens and two cameras is shown in the fig.3. In this case, on each screen, two beam profiles are created, corresponding to different sections of the main beam. Fig. 4 shows the images of 4 spots, in 4 planes of the beam, obtained by the SIEPCH‑2 measuring device. Each profile is processed using the software of the SIEPCH‑2 meter separately. Parallel registration of the beam profile in four planes makes it possible to estimate not only the variation of both components of the divergence, but also the change in the beam quality in time, determined by the parameters M2 and BPP, both for a continuous mode of operation, and for a pulse-periodic operation. It is obvious, that the focal lengths of thermal lenses, determined by the above formulas, are very large and therefore inconvenient for constructing their time dependences FT (t) (all begin with infinity). Much more evident in this sense are the time dependences of the corresponding optical force D(t) = 1 / FT(t) of thermal lenses, which make it possible to estimate, in addition, the rate of increase of the optical force at different time intervals of generation. In our experience, it is most convenient to evaluate the optical power of thermal lenses in reverse kilometers (km–1). As an illustration of the measurement procedure described above, Fig. 5 shows one of the realizations of the change in the optical power of a positive thermal lens arising on the output optics of a single-mode fiber laser with a power of 1.7 kW. At an initial section of about 20 seconds, the optical force increases with an average speed of about 0.5 km–1/s. Then, until the end of the 110-second start, it remains approximately at the same level D (t) ≈ 10 km‑1 (the equivalent focal length of the thermal lens FT(t) ≈ 100 m). In a multimode fiber laser with a power of 5 kW, the time variation of the geometric divergence, according to our observations, is much less pronounced. But there is a noticeable (about 7%) increase in physical divergence during the first seconds of generation. Thus, the parallel measurement of the profile of the focused beam in four sections makes it possible to control the deformation of the waist of the focused beam in time. In this case, the width of the waist is correlated with changes in the physical divergence, and the waist shifts along the beam axis uniquely determine the changes in geometric divergence.