Calculation and analysis of the laser beam field distribution formed by a real optical system

When designing high-quality optical systems, it is necessary to calculate the real field distribution of the output beam. In this work, the solution of this problem by the method of the scalar theory of diffraction is considered. For calculate the diffraction integral and visualize the results, the software package Mathematica (Wolfram Mathematica) is used. Aberration analysis of laser optical systems and simulation in the program Mathematica of the output field distribution showed a significant distortion of the field distribution and deviation of the spatial beam parameters from the ideal one, described by analytic expressions.


Introduction
Nowadays a large number of lasers is produced, and they are generously used in science, technology, industry and other sectors of the national economy [1][2][3][4][5][6][7][8][9]. However, during the process of developing laser optoelectronic devices it is a common situation, when opticians take into account only the wavelength, divergence and beam diameter at the output of the laser. These parameters do not fully characterize the laser radiation generated by the optical resonator. This is also shown by the fact that in laser's datasheet the optical scheme of the resonator and its design parameters, which determine the full set of output radiation parameters, are absent, with rare exception. Nevertheless, knowledge of entire laser radiation parameters is necessary for proper calculation of the forming laser optical system (LOS). The proper LOS calculation, as practice shows, allows to improve the quality of the formed beam and the technical characteristics of the developed laser devices and systems by designing an optical unit with smaller dimensions, mass, using a laser radiation source of lower power, i.e. to reduce power consumption [10][11][12][13][14].
Usually laser beams are formed by optical systems consisting of lens, mirrors, diffraction optical elements (DOE) and other components [11,12,15,16]. The paper [16] presents the results of the research of laser radiation focusing using different types of axial-symmetric diaphragms and binaryphase plate. The problem of calculating the amplitude, phase and intensity of the field near the focus in the presence of primary aberrations of the optical system for the case of uniform distribution of the field amplitude on the input pupil is considered in the paper [17].
A number of corresponding methods have been developed to calculate optical systems of different types [11,12,[18][19][20]. Diffraction methods of calculation are widely used in optics. Also, diffraction methods allow to conduct a complete analysis of the properties of laser radiation formed by optical resonators [10,11]. In the development of high-quality LOS, the final assessment of the quality of the beam formed by the optical system should also be carried out by diffraction methods. Since the numerical solution of the diffraction integral is a rather laborious task, in engineering practice approximate methods of calculation are important, which, under reasonable assumptions, can significantly simplify the problem. This approach allows to obtain approximate analytical expressions for estimated calculation.
This paper presents approximate expressions that allow to analyze aberration distortions of a stable laser resonator beam at the LOS output, as well as the results of direct calculation of the diffraction integral for calculating the output distribution of the complex field amplitude and the parameters of the formed laser beam.

Main calculation expressions
The method of calculation of complex amplitude distribution of the laser radiation field at the output of the optical system, which takes into account its aberrations, is based on the scalar diffraction theory. According to the Rayleigh-Sommerfeld formula, the distribution of the complex field amplitude    (1) where  is the distribution of the complex field amplitude of the laser beam output from the reference sphere  ,  is the wavelength of the radiation,  The distribution of the complex amplitude of the Hermite-Gauss beam field on the output reference sphere after the real LOS with accuracy to a constant phase factor is determined by the following expression [23]: Here a the distribution of the field amplitude of the Hermite-Gaussian beam with mn TEM mode at the input LOS in the plane 11 xy,   the coefficient, that characterizes the ratio of the laser beam sizes in the planes  and 11 xy. Wave aberration function of the real LOS from the transverse coordinates on the output reference sphere has the form ( Fig. 1): ,,    and  are the coefficients of decomposition, that characterize, respectively, the defocus and aberration of the LOS of the 3rd, 5th and 7th orders. These expressions allow us to carry out a numerical simulation of the laser beam propagation through the real optical system and to plot the functions of distortion of the amplitude and phase distribution of the field, as well as to calculate the deviation of the spatial parameters of the real beam from the perfect ones, formed by ideal (non-aberration) LOS. The methods of aberration distortion calculation of the laser and incoherent radiation optical systems should be recognized due to the following: firstly, the laser beam is described only by so-called ray bundle [12,24]; secondly, the radii of curvature of the input and output reference spheres are calculated by the formulae of laser optics [12] through the paraxial parameters of the input and output beams respectively: . Here c z is the Rayleigh length of the output beam of the ideal LOS, w s is the position of the output waist relative to the last surface of the LOS. The expressions for R  and R   are given according to the sign rule, accepted in the laser optics.
To obtain an approximate analytical expression for the complex field amplitude distribution of the beam transformed by the LOS in the diffraction integral (1) it is necessary to make the following assumptions. Firstly, the angular coefficient is considered to be equal to one in the denominator of the integrand AP PQ rz  . Secondly, the phase multiplier PQ r should be represented as Taylor series, taking into account the terms to the eighth order of smallness, and then the economization of obtained expressions should be held with the help of Chebyshev polynomials on the canonical interval [0;+1] [23,24]. In this case the aperture of the LOS component is considered to be more (2.5...3.0 times) than the size of the beam. As a result, we can distinguish the main part of the aberration terms (quadratic), which takes into account 1/2, 15/32, 7/16 of the LOS aberrations of the 3rd, 5th and 7th orders [23]: 2h is the light diameter of the last refractive surface of the LOS. After the analytical integration of the diffraction integral, the field distribution and the generalized dependences of the laser beam which take into account the LOS aberrations are obtained. Since they take into account main part of the LOS aberrations, they describe the aberration of Hermite-and Laguerre-Gaussian beams, respectively. The dependences for the envelope of an arbitrary transverse mode mn TEM ( ab mn h ) and the radius of curvature of the wave front ( ab R   ) on the analysis plane AP z position can be described by equation [23]: Here mn  is the coefficient that determines the increase of the spot size of the higher transverse mode mn TEM compared to the main one 00 TEM that calculated by the method of moments [22]. In the above relations, the LOS aberrations are included in the parameter A as a coefficient eq  for the Chebyshev polynomial of the 1st order:      In the relations above, the values max h and max h characterize the maximum size of the region (its radius) on the output reference sphere  and in the plane of the output field analysis, in which the vast majority of the given transverse beam mode energy is concentrated.
It follows from equations (4) that the envelope of the beam transformed by the LOS with aberrations is no longer a second-order surface (rotational hyperboloid), as in the case of paraxial approximation.
The main spatial parameters of the aberration beam, namely: the position of the waist and its size, angular divergence) and the beam quality parameter 2 M can be calculated using the following formulae [ According to these formulae (6) for the diameter and position of the output waist the value A , taking into account the LOS aberrations, is in the denominator, and for the angular divergencein the numerator. Therefore, the presence of aberrations has different effects on the beam parameters at the LOS output. Rear waist shift in real LOSs is usually described in laser technology by focal shift factor FSF wc zz 


, where w z  is waist shift relatively to the ideal LOS waist position.  The radius and the position of the beam waist determined by the calculation of the diffraction integral (1) for the ideal LOS in Wolfram Mathematica software package are equal to 4.50 µm and 11.78 mm respectively, i.e. they coincide with the results of the calculation according to analytical formulae.

Real LOS distortions analysis example
To estimate the distortion of the output beam field by the ray bundle method, the wave aberration was calculated [23] (fig. 2, 3). Wave aberration was calculated with LOS depreciation factor equal to 2.5.
The    The radius and the position and output waist, as well as the beam parameter 2 M calculated by formulae (5), are 4.52 µm, 11.85 mm and 1.003 respectively. The radius and the position of the beam waist determined by the calculation of the diffraction integral (1) in the Wolfram Mathematica, taking into account (2) and (3), have values of 4.55 µm and 11.81 mm (Fig. 4).   (5), is 11.71 mm, and as a result of the calculation of the diffraction integral -11.73 mm. When the expansion coefficients of the LOS wave aberration function have different signs, the unambiguous conclusion about the direction of displacement of the output waist cannot be made.

Discussion
Summing up the results of aberrational and diffractive distortions analysis made by the means of diffraction integral numerical calculation in the package Wolfram Mathematica it is important to note the following:  field amplitude distribution after the real LOS is close enough to the Gaussian;  real and paraxial beam parameters are different;  real beam envelope has longitudinal asymmetry. Concerning the third clause we want to emphasize the following. In laser technology the confocal parameter, also known as Rayleigh length, is widely used. Its double value is called «waist length». Rayleigh length is defined as the distance between the waist and beam cross-section with times larger diameter, than in the waist. Owing to the real beam envelope distortion, these distances in the front of and behind the waist are different. For the considered LOS the front one equals 118 µm, and the rear one 93.6 µm. FSF parameter equals 0.32. The analysis carried out has shown, that the particular values of these distances depend on the LOS aberrations. The LOS with different spatial parameters can form the beam with either longer front or longer rear confocal parameter.

Conclusion
The analysis of the laser beam propagation through the optical system taking into account its aberrations has shown that the calculation of the field distribution and the beam spatial parameters by approximate formulae is consistent with the results of direct calculation of the diffraction integral.
The difference is caused by taking into account all LOS aberrations and free space, as well as increased accuracy of calculating the diffraction integral in the Wolfram Mathematica software package. In addition, the output waist displacement relative to its paraxial position and the deviation of the spatial parameters of the beam are determined by the wave aberration of the LOS, namely its expansion coefficients. The results we have obtained, should be taken into account when designing laser optics for precision laser technologies.