Study On The Propagation Properties, Coherent And Incoherent Combinations Of Nonparaxial Beams

With the development of laser technology, espeialiy for the wide application of diode lasers and practical technology, there exist optical beams of very small spot size and large far-field divergence angle, for which the paraxial theory is invalid, and lots of study has been intrigued for nonparaxial propagation. The first systematic approach to nonparaxial propagation goes back to pioneering work of Lax et al. Following this pioneering work, other approaches, such as perturbation power series method, angular spectrum representation, transition operators, and virtual source method, developed by Davis, Agrawal, Pattanayak, Couture and Belanger, Takenaka, and Wünsche et al., are used to study nonparaxial propagation of light beams.Beam combination provides a useful tool for scaling diode arrays, CO₂ and HF/DF chemical lasers as well as fiber lasers etc. up to high-power level. Up to now, a variety of linear, rectangular and radial beam combination schemes have been proposed and studied both theoretically and experimentally, but are restricted within the paraxial regime. However, the paraxial approximation fails for beams such as diode-laser beams with large divergence angle and/or small beam width comparable with the wavelength, so the nonparaxial approach has to be taken into consideration.In this dissertation, some studies on the propagation properties, coherent, and incoherent combinations of nonparaxial beams have been done. They are summarized as follows: Stating from the Rayleigh-Sommerfeld diffraction integral, the propagation expressions in free space for the scalar nonparaxial Bessel-Gaussian beams, Dark-hollow Gaussian beams(DHGBs), Phase flipped Gaussian(PFG) beams are derived and the corresponding far-field results are also presented for the first time. By using these expressions, which are in consistence with the results under the paraxial approximation, the propagation properties of these different beams are studied. Comparison of the results of different approaches shows that the method of using the Rayleigh-Sommerfeld integrals possesses obvious advantages over the perturbation series method. The methods used in this dissertation are relatively simple and efficient in finding the analytical expressions for the nonparaxial beams, which are more tractable in analyzing the propagation properties of the nonparaxial beams and the divergence of perturbation series solution is avoided.The phase-flipped Gaussian(PFG) beam is proposed for the first time. Starting from the Rayleigh-Sommerfeld diffraction integral, the recurrence propagation expressions for nonparaxial PFHG beams have been derived and used to study the propagation properties of PFHG beams in free space and through a knife edge and an aperture for the first time. The propagation of paraxial PFHG beams and PFG beams and nonparaxial PFG beams is treated as special cases of nonparaxial PFHG beams. Numerical examples are given to illustrate the theoretical results. It is shown that the /-parameter, mode indices, offsetting distance, and truncation parameter affect the nonparaxial propagation behavior of PFHG beams. The propagation of paraxial PFHG beams and PFG beams and nonparaxial PFG beams can be regarded as special cases of our general results and treated in a unique way. Under certain conditions, the nonparaxial approach described in this paper has to be taken into consideration. The aperture effect is negligible atδ≥2.1 for the PFHG beam in through an aperture.By using the vectorial Rayleigh-Sommerfeld integrals, the propagation expressions in free space for the vectorial nonparaxial cosh-Gaussian(ChG) beams, vectorial nonparaxial off-axis Gaussian beams, and vectorial nonparaxial four-petal Gaussian beams(FPGBs) are derived and the corresponding far-field results are also presented for the first time. By using these expressions, which are in consistence with the results under the paraxial approximation, the propagation properties of these different beams are studied. It is shown that the field of nonparaxial beams is vectorial in nature, so both the longitudinal and the transversal components of the field should be taken into account. The nonparaxiality and vectoriality of the beams increases with the increasing parameter f.The propagation equation of vectorial nonparaxial cosh-Gaussian(ChG) beams and vectorial nonparaxial off-axis Gaussian beams in the presence of an aperture are derived and expressed in a closed form for the first time. The on-axis, far-field and paraxial cases are studied as special cases. As can be seen, the aperture has strong diffraction effects on the propagation field. The diffraction field of aperture is also vectorial in nature; apart from the transversal component, the longitudinal component of the diffraction field becomes nonnegligible and begins to increase with the decreasing of the dimension of the aperture. The existence of an aperture enhances the nonparaxiality and vectorial properties of the diffraction field, which depends on the parameter f and the truncation parameterδ. The two f-parameters f_x and f_y, and two truncation parametersδ_x andδ_y have to be introduced to describe properties of the diffraction field by a rectangular aperture. For the vectorial nonparaxial ChG beams, only when the truncation parameterδ>2.1, can the effect of the aperture be neglected. For the vectorial nonparaxial off-axis Gaussian beams, the/parameter still plays a key role in determining the beam nonparaxiality, but the truncation parameterδand off-axis parameters x_d(y_d) additionally affect the beam nonparaxial evolution behavior, the unsymmetry of(x, y)_max is caused by the different contribution of I_z(x, y, z) in the x and y directions.The partial coherence is an inherent property of some optical sources(or beams), so it is of great theoretical and practical importance to consider the effects of coherence of light on the nonparaxiality of beams. This dissertation expands the concept of nonparaxiality from completely coherent beams to partially coherent beams. Based on the Rayleigh-Sommerfeld diffraction integrals, the propagation properties of the partially coherent nonparaxial J₀-correlated Schell-Model(JSM) beams, partially coherent nonparaxial modified Bessel-Gauss(MBG) beams and partially coherent nonparaxial Hermite-Gaussian(HG) have been studied for the first time. It is shown that besides the parameter f a new parameter f_σis needed to describe the nonparaxiality of the partially coherent beams. For the partially coherent nonparaxial JSM beams, the paraxial approximation is applicable, if the parameters/ andσare small, e.g.f=0.01,σ=2.0, otherwise, the nonparaxial approach should be applied. For the partially coherent nonparaxial MBG beams, the intensity and spectral degree of coherence of nonparaxial MBG beams depends on the order m,ξand f parameters, whereas their cross-spectral density function is additionally dependent on the orientation△φ. The nonparaxial approach has to be used once the f parameter is no small. For the partially coherent nonparaxial HG beams, both f and f_σparameters determine their nonparaxility of partially coherent TEM_mn-mode HG beams. The paraxial approximation is allowable if the f and f_σparameters are small enough. The values of f and f_σparameters taken for the nonparaxial case are dependent on the mode indices m, n.Taking the two-dimensional(2D) off-axis astigmatic Gaussian beam combination as an example, the concept of the beam combination is extended to the nonparaxial regime for the first time. The analytical propagation expressions for coherent and incoherent combinations of 2D nonparaxical off-axis astigmatic Gaussian beams have been derived and utilized for studying the coherent and incoherent beam combinations beyond the paraxial approximation for the first time. It has been found that generally, the intensity distributions of the resulting beam are dependent on the combination scheme(coherent or incoherent combination), and beam parameters including the waist widths w_0x, w_xy(or equally, f-parameters), separate distances x_d, y_d, and numbers of beamlets m, n, but the far-field intensity distributions in the x and y directions for the incoherent combination are independent of the separate distances. The paraxial results can be regarded as special cases of the nonparaxial ones, and only for the small f-parameter, e.g.f=0.01, is the paraxial approximation valid. Otherwise, the nonparaxial approach should be taken into consideration.In conclusion, this dissertation explicates the use of Rayleigh-Sommerfeld integrals in studying the propagation of nonparaxial beams. The concept of the beam combination is extended to the nonparaxial regime. The propagation properties for coherent and incoherent combinations of 2D nonparaxial off-axis astigmatic Gaussian beams with rectangular geometry are studied. Using the methods developed, this dissertation expands the study of nonparaxial theory and solves many intractable problems of the existing theory, such as the nonparaxial effects of the optical aperture, the propagation of the partial coherent nonparaxial beams and coherent and incoherent combinations of nonparaxial beams.