Calculation methods

Calculation of the radius of the TEM00 mode of the laser cavity (the lowest, Gaussian mode) is carried out by the well-known method of ABCD ray matrices. Briefly, the essence of the method is as follows. A 2x2 square matrix is assigned to each optical element of the laser cavity and the gaps of free space in it. Examples of matrices are given in the table below.

Optical element Matrix
Mirror with radius of curvature R
Thin lens with focal length f
The gap of a medium with a refractive index n and a length L
General view of the matrix ABCD

The matrix allows expressing the ray parameters (the position of the ray relative to the optical axis and the angle of inclination of the ray to the optical axis) at the output of the optical element in terms of the ray parameters at the input. If you need to find the output ray on base of the input ray for a multi-element system, then you need to multiply the matrices of elements in order from the output to the input of the optical system.

The radius of curvature of a spherical beam obeys the law ABCD:

(R1 and R2 – radius of curvature of the input and output beam, respectively).

The so-called complex parameter of the Gaussian beam obeys the same law, which makes it possible to calculate the parameters of the Gaussian beam by the matrix method.

The radius of the TEM00 mode according to the level 1/e of the amplitude of the electric field strength (i.e., according to the level 1/e2 of maximum intensity) in any reference plane is determined through the elements of the matrix ABCD, written for a complete bypass of the laser cavity relative to this plane, by the formula:

(λ - radiation wavelength).

More detailed information on the method can be obtained in the literature on laser technology.

When calculating the beam divergence in the application, it's assumed that the output mirror (if not flat) has a substrate compensating the spherical component of the divergence, i.e. the beam waist is located directly behind the output mirror. In this case, the divergence of the TEM00 mode (in terms of intensity level 1/e2) is expressed by the formula:

(ωout – TEM00 mode radius at the output mirror).

When calculating the beam as a whole (with higher modes), it is assumed that it occupies the entire aperture of the lens (we assume that the lens is induced in the active element and its diameter is equal to the diameter of the generation region in the element), and the aperture of the mirrors obviously exceeds the size of the field on them. In this case, in any transverse section of the beam, its size is equal to the size of the TEM00 mode in this section, multiplied by the multimode factor. The multimode factor is calculated as the ratio of the lens radius to the radius of TEM00 mode in it. The divergence of the entire beam is assumed to be equal to the divergence of the TEM00 mode multiplied by the multimode factor.

It is assumed that the radius of the entire beam is not less than the radius of the TEM00 mode.

Focusing of power is defined as the inverse square of the ratio of the field radius in the calculated laser cavity transverse section to the field radius in the lens.

The aperture losses in the lens for the Hermite-Gaussian modes are calculated for a round trip of the laser cavity. The transmission through the lens in each direction along the laser cavity optical axis is found as a fraction of the mode energy localized inside the lens aperture of the total energy. For the first six modes (00…22) the transmission through the lens is determined by the expressions:

(R – radius of lens, ω – radius of TEM00 mode in lens).

Thus, aperture losses of mn-mode (in percentage terms):

It should be noted that the method for calculating higher modes is very approximate. Therefore, the results of the calculation of higher modes should be considered only estimates.

As aperture losses increase, the role of aperture effects increases, and in this case, more accurate diffraction methods for calculation should be used. However, even in this case, the calculation of the aperture losses is useful, because allows you to evaluate the "chances of survival" of specific modes.

It should also be noted that the paraxial approximation is used in the calculations, so the results are applicable only for paraxial beams, with a not too large divergence. The author of the application met the such estimate: for the max. angles of rays not exceeding 100 mrad (approx. 60) to the optical axis the calculation error in the paraxial approximation is less than 1%.


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