1. Introduction

Optical vario- or zoom systems is a class of optical systems for varying their optical characteristics (focal length, magnification, angular field, etc.) due to a change in the scheme configuration. The advent of lasers, the development of laser engineering and technology have led to the emergence of a subclass of optical variosystems – laser variosystems (LVSs). Variosystems are used in optical devices and apparatuses: video- and photoequipment, mobile phones, projectors [1–4]; in technologies: laser processing of materials, additive technologies [5–9] and other fields [10,11].

Different methods are used to develop variosystems. In the classical method, components with a fixed focal length are usually moved along the optical axis [12–20]. There are variosystems that use components moving across the optical axis, but their use is limited by a complex freeform surface and a small zoom range [21–23]. The movement of components in modern systems is carried out mainly using a high-precision electromechanical drives. The classical method is now widely used, because it has few limitations. Basically, limitations are related to the dimensions of the system, i.e. a large component displacement or a complicated law of component movement.

The use of tunable (liquid) lenses for constructing variosystems becomes popular [24–26]. However, commercially available liquid lenses still have some limitations [27,28]. Liquid lenses based on an elastic membrane with a large diameter (> 20 mm) are manually controlled, which reduces their speed, or require an electromechanical drive. The technology of liquid lenses based on the electro-wetting effect allows to obtain small light diameters – about 3 mm. The limitation of the practical use of liquid lenses is related to their small aperture.

It seems promising the combined method of LVS design, which consists in the use of tunable lenses in combination with their movement or the movement of conventional fixed focus lenses. This method was used to design a mobile phone camera [29].

The properties of laser radiation [30–35] do not allow the use of classical optics methods for calculating laser optical systems. Special techniques are required for their calculation and development.

The synthesis of a combined LVS includes two main stages: structural-dimensional synthesis and aberration synthesis. At the first stage, the structure, dimensional parameters, laws of moving or changing the optical powers of the LVS components, at which a laser beam with the required parameters are formed, are determined. At the stage of structural-dimensional synthesis the components are considered ideal (non-aberrational). The design parameters of the LVS are determined at the stage of aberration synthesis [20,34,35].

The problem of structural-dimensional synthesis of a LVS for the longitudinal movement of a Gaussian beam waist with a constant diameter is considered in the paper. LVS schemes with the combined method of optical characteristics varying are analyzed.

2. Transformation of Gaussian laser beams by an ideal optical system

For calculating the spatial parameters of the Gaussian beam formed by a cavity and an optical system, the Kogelnik ABCD ray transfer matrix method is widely used [30–34]. The use of the matrix method for multi-component laser optical systems leads to cumbersome expressions, which complicates the task of their development. Despite the development of computational tools and numerical methods, the task of developing methods for analytical synthesis and analysis of laser optical systems remains relevant.

We obtain expressions for the analysis and dimensional synthesis of optical systems that form Gaussian laser beams (TEM₀₀ mode). These expressions can be effectively used to develop methods for the synthesis of LVSs using different principles (including combined LVSs that provide longitudinal movement of the output Gaussian beam waist, keeping its diameter constant).

The beam at the input of the optical system is characterized by the following parameters (Fig. 1): radiation wavelength $\def\upbeta{\unicode[times]{x3B2}}\lambda $, radiation power, waist diameter $2{h_\textrm{w}}$, waist position relative to the front principal plane of the optical system ${s_\textrm{w}}$, Rayleigh length ${z_\textrm{c}}$ or confocal parameter $2{z_\textrm{c}}$, angular divergence $2\theta$, beam quality factor ${M^2}$. Wherein the spatial parameters of the Gaussian beam satisfy the invariant [33,34,36]: $2{h_\textrm{w}}2\theta = 4{{h_\textrm{w}^2} \mathord{\left/ {\vphantom {{h_\textrm{w}^2} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = \textrm{const} = 4{M^2}{\lambda \mathord{\left/ {\vphantom {\lambda \pi }} \right.} \pi }$. Thus, using the known parameters of the beam, it is possible to calculate the distribution of the complex field amplitude in any section at the input of the optical system [30–34].

 

Fig. 1. Transformation of a Gaussian laser beam by an optical system: H and $H^{\prime}$ are the front and rear principal points, F and $F^{\prime}$ are the front and rear focal points, $\textrm{W}$ is the beam waist, $2{h_F}$ is the diameter of the transformed Gaussian beam in the front focal plane, ${s^{\prime}_\textrm{w}}$ is the distance from the back principal point to the output waist.

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We will solve the problem within Gaussian optics for an ideal optical system using the scalar theory of diffraction (Fig. 1). The transverse distributions of the complex field amplitude $\psi $ in the conjugate planes of the optical system are related by the expression [32,37]:

According to Eq. (1), in the conjugate planes of the optical system, power density distributions M are similar up to a scale factor $\upbeta $: $M^{\prime}({x^{\prime},y^{\prime}} )= \frac{1}{{|\upbeta |}}M\left( {\frac{{x^{\prime}}}{\upbeta },\frac{{y^{\prime}}}{\upbeta }} \right)$, and the phase distribution $\phi$ accurate up to a constant factor is: $\phi ^{\prime}({x^{\prime},y^{\prime}} )= \phi \left( {\frac{{x^{\prime}}}{\upbeta },\frac{{y^{\prime}}}{\upbeta }} \right) - \frac{k}{{2\upbeta f^{\prime}}}({{{x^{\prime}}^2} + {{y^{\prime}}^2}} )$. Consequently, the transformed beam remains Gaussian and can be characterized by the same dependences as the input beam, but with different spatial parameters.

In the conjugate planes of the optical system P and P’, the diameters of the Gaussian beam $2h$ at the same level of the radiation power density and the wavefront curvature radius ${R_\Phi }$ of the beam are determined through the transverse magnification $\upbeta $ by the following expressions (Fig. 1):

Let us consider an arbitrary cross section of a beam transformed by an optical system with a coordinate $z^{\prime}$ relative to the rear focal plane. A conjugate plane with a coordinate $z({z^{\prime}} )={-} {{{{f^{\prime}}^2}} \mathord{\left/ {\vphantom {{{{f^{\prime}}^2}} {z^{\prime}}}} \right.} {z^{\prime}}}$ will correspond to this cross section at the optical system input, and the magnification in these planes is $\upbeta ({z^{\prime}} )={-} {{z^{\prime}} \mathord{\left/ {\vphantom {{z^{\prime}} {f^{\prime}}}} \right.} {f^{\prime}}}$. Then, taking into account Eqs. (2) and (3), we obtain the dependences for the envelope diameter and the wavefront curvature radius of the transformed beam in the cross section $z^{\prime}$:

The transformation of a Gaussian laser beam by an optical system can be characterized by a longitudinal magnification for the near zone ${\alpha _G}$ and a transverse magnification in waists ${\upbeta _G}$. They are determined by the focal length of the optical system, the Rayleigh length of the transformed beam and the position of its waist with the following expressions:

After transformations of Eq. (4) and Eq. (5), taking into account Eqs. (6)–(10), they can be written in the traditional form:

Table 1. Formulas for calculating the parameters of a Gaussian beam at the output of two- and three-component optical systems.

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Let us define the length of the laser optical system as the distance from the input waist to the output one: $L ={-} {s_{\textrm{w1}}} + \sum\limits_{j = 1}^{m - 1} {{d_j}} + \sum\limits_{j = 1}^m {{\Delta _{{H_j}{{H^{\prime}}_j}}}} + {s^{\prime}_{\textrm{w}m}}$, where ${\Delta _{{H_j}{{H^{\prime}}_j}}}$ is the distance between the principal planes of the $j$-th component.

Let us analyze the dependence of the positions of the beam waists at the input and output of the optical system and the dependence of the position of the input waist on the focal length of the optical system when transforming a Gaussian beam with magnification ${\alpha _G} = \textrm{const}$ (Eq. (10)). Firstly, the output waist is not an image of the of the input beam waist, because the waists at the input and output of the optical system are not in optically conjugate planes as for incoherent homocentric radiation. The position of the waist and the spatial parameters of the transformed beam are determined by the formulas of laser optics (Eq. (10)). Secondly, the position of the output waist relative to the rear focal plane is in the range $[{{{ - {{f^{\prime}}^2}} \mathord{\left/ {\vphantom {{ - {{f^{\prime}}^2}} {2{z_\textrm{c}}}}} \right.} {2{z_\textrm{c}}}};{{{{f^{\prime}}^2}} \mathord{\left/ {\vphantom {{{{f^{\prime}}^2}} {2{z_\textrm{c}}}}} \right.} {2{z_\textrm{c}}}}} ]$ (Fig. 2(a)), (i).e. the optical system forms a waist of a Gaussian beam at a finite distance, in contrast to the transformation of incoherent homocentric radiation, when the image position can be at infinity. Thirdly, to form a Gaussian laser beam, an optical system (lens) should be used, the focal length of which is greater than the minimum (Fig. 2(b)). This minimum focal length is determined by the spatial parameters of the input and output beams: ${f^{\prime}_{\min }} = {z_\textrm{c}}\sqrt {{\alpha _G}} = {{{h_\textrm{w}}} \mathord{\left/ {\vphantom {{{h_\textrm{w}}} {\theta^{\prime}}}} \right.} {\theta ^{\prime}}} = {{{{h^{\prime}_\textrm{w}}}} \mathord{\left/ {\vphantom {{{{h^{\prime}_\textrm{w}}}} \theta }} \right.} \theta } = \sqrt {{z_\textrm{c}}{{z^{\prime}_\textrm{c}}}}$.

 

Fig. 2. Transformation of a Gaussian beam by an optical system with a given magnification ${\alpha _G} = \textrm{const}$: a – dependence of the position of the waists at the optical system input and output relative to the focal planes, b – dependence of the position of the input waist relative to the front focal plane from the optical system focal length.

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Let us use the obtained dependences and expressions to develop a synthesis technique for LVS that provides longitudinal movement of the output Gaussian beam waist while keeping its diameter constant.

3. Method for calculating laser variosystems

An analysis of the Gaussian beam transformation by a laser optical system yields the following result: the angular divergence of the Gaussian beam at the optical system output is equal to $2{\theta ^{\prime}_m} = {{2{h_{{F_m}}}} \mathord{\left/ {\vphantom {{2{h_{{F_m}}}} {|{{{f^{\prime}}_m}} |}}} \right.} {|{{{f^{\prime}}_m}} |}}$, where $2{h_{{F_m}}}$ is the beam diameter in the front focal plane of the last $m$-th component of the optical system, ${f^{\prime}_m}$ is the focal length of the $m$-th component of the optical system. Since $2{h^{\prime}_{\textrm{w}m}}2{\theta ^{\prime}_m} = 4{M^2}{\lambda \mathord{\left/ {\vphantom {\lambda \pi }} \right.} \pi }$ (see Eq. (11)), longitudinal movement of the output beam waist while maintaining its diameter constant is possible under the following condition:

According to [40], in which the synthesis of a two-component LVS with a fixed focal length of the components and their movement is considered for the formation of a Gaussian beam of constant diameter at various distances from the input waist (of a laser), the solution of this problem is possible when changing only two design parameters of the LVS. Using Eqs. (6)–(14), we obtain expressions for the dimensional calculation of one- and two-component LVSs. In this case, the position of the waist is preferably to determine not from the focal planes of the lens F and $F^{\prime}$, but from its principal planes H and $H^{\prime}$:

The solution of the dimensional calculation of an LVS is its structural scheme with a set of dimensional parameters and laws of their change to ensure the required movement of the output waist of constant size when all the restrictions imposed on these parameters and the law are met.

The indicated restrictions are very diverse and are usually imposed on the longitudinal and transverse distances and sizes of the LVS (length, waists position, etc.), the apertures of the components, the maximum allowable lens movement, the number of components, and other parameters. The LVS scheme must be physically feasible, i.e. all distances between the previous and subsequent components of the scheme must be positive, the focal lengths of the tunable liquid lenses must be within the physically realizable range of focal lengths for a given lens design. Moreover, it is namely the restrictions that are the main reason for the solution absence for the dimensional synthesis of the considered LVSs. Nevertheless, there can be a lot of solutions, and to determine the best solution at the stage of dimensional synthesis, one should use the objective function containing the most important structural-dimensional parameters of the LVS.

3.1. One-component laser variosystems

In this case, it is necessary to use a tunable lens moving along the optical axis (Fig. 3). Equation (13) taking into account Eq. (9) and Eq. (15) has the following form:

 

Fig. 3. A single-component LVS for the formation of a Gaussian beam waist of constant diameter at various distances from the input waist.

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The length of a single-component laser optical system is determined by the expression (see Fig. 3): $L ={-} {s_\textrm{w}}({{\alpha_{G\textrm{ req}}} + 1} )- f^{\prime}({{\alpha_{G\textrm{ req}}} - 1} ).$ This expression, considering the expression for $f^{\prime}$ after transformations, allows one to obtain the dependence of the length of a single-component LVS on the distance between the lens and the input waist:

The solution to the dimensional problem of a single-component LVS is possible only if the following conditions are met: 1) the focal length of the lens $|{f^{\prime}} |> {f^{\prime}_{\min }} = {z_\textrm{c}}\sqrt {{\alpha _{G\textrm{ req}}}}$, 2) the position of the input waist relative to the lens $|{{s_\textrm{w}}} |> {s_{\textrm{w min}}} = {z_\textrm{c}}\sqrt {{\alpha _{G\textrm{ req}}} - 1}$, 3) the position of the output waist relative to the lens $|{{{s^{\prime}_\textrm{w}}}} |> {s^{\prime}_{\textrm{w min}}} = {z_\textrm{c}}\sqrt {{\alpha _{G\textrm{ req}}}({1 - {\alpha_{G\textrm{ req}}}} )}$. Thus when ${\alpha _{G\textrm{ req}}} < 1$ there is a restriction on the position of the output waist, and when ${\alpha _{G\textrm{ req}}} > 1$ – of the input waist. Figure 4 shows the boundary curves of the normalized design parameters of a single-component LVS from the required longitudinal magnification, where a solution to the dimensional problem exists in the shaded area.

 

Fig. 4. Boundary curves of normalized design parameters of a single-component LVS from the required longitudinal magnification: a – minimum focal length of the lens, b – distance from the waist to the lens, c – distance from the lens to the output waist.

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Note that restrictions on the design parameters of a single-component LVS also occur during the dimensional synthesis of multi-component LVS, for the calculation of which in these formulas it is necessary to substitute the Rayleigh length of the beam at the input of the corresponding component, and instead of the required magnification in the variosystem – magnification the component.

Let us illustrate the application of the obtained formulas for solving the dimensional problem of a single-component LVS (Fig. 5). Given the known ${z_\textrm{c}}$, ${\alpha _{G\textrm{ req}}}$, maximum and minimum boundaries of the LVS length ${L_{\max }}$ and ${L_{\min }}$, the range of movement of the waist $\Delta L = {L_{\max }} - {L_{\min }}$, using Eq. (18), we determine the range of the position of the input waist $\Delta {s_\textrm{w}} = {s_{w\max }} - {s_{w\min }}$. Further, using Eq. (17) for the required range of lens movement $\Delta {s_\textrm{w}}$, we determine the range of its focal length variation.

 

Fig. 5. Dependences of the normalized length ${L \mathord{\left/ {\vphantom {L {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = \bar{L}$ (a, b) and the normalized focal length ${{f^{\prime}} \mathord{\left/ {\vphantom {{f^{\prime}} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = \bar{f^{\prime}}$ (c, d) of a single-component LVS on the normalized distance between the lens and the input waist ${{{s_\textrm{w}}} \mathord{\left/ {\vphantom {{{s_\textrm{w}}} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = {\bar{s}_\textrm{w}}$: a, c – ${\alpha _{G\textrm{ req}}} < 1$, b, d – ${\alpha _{G\textrm{ req}}} > 1$ (normalized range of the LVS length ${{\Delta L} \mathord{\left/ {\vphantom {{\Delta L} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = \Delta \bar{L}$, normalized range of the focal length LVS ${{\Delta f^{\prime}} \mathord{\left/ {\vphantom {{\Delta f^{\prime}} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = \Delta \bar{f^{\prime}}$, normalized range of distance between the lens and the input waist ${{\Delta {s_\textrm{w}}} \mathord{\left/ {\vphantom {{\Delta {s_\textrm{w}}} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = \Delta {\bar{s}_\textrm{w}}$).

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It can be seen from the graphics in Fig. 5 that for a given range of LVS length change $\Delta L$, there is one range of the input waist position relative to the lens – for ${\alpha _{G\textrm{ req}}} < 1$ with the virtual and real waists (Fig. 5(a)) and for ${\alpha _{G\textrm{ req}}} > 1$ with the virtual waist (Fig. 5(b)), and two ranges of the lens focal length (Fig. 5(c) and (d)) variation.

When moving the lens, the diameter $2{h_L}$ of the Gaussian beam and its light diameter $2{h_{\textrm{ca}}}$ change in accordance with the following relationships:

The f-number is calculated by the formula: $Nd = {{f^{\prime}} \mathord{\left/ {\vphantom {{f^{\prime}} {2{h_{\textrm{ca}}}}}} \right.} {2{h_{\textrm{ca}}}}}$.

When moving the lens, the position of the input waist changes as follows: ${s_\textrm{w}}({{\delta_L}} )= s_\textrm{w}^{(0 )} - {\delta _L}$. Here $s_\textrm{w}^{(0 )}$ and ${s_\textrm{w}}$ are the initial and current positions of the waist of the transformed beam relative to the lens, ${\delta _L}$ is the movement of the lens: ${\delta _L} < 0$ – to the left relative to the initial position, ${\delta _L} > 0$ – to the right relative to the initial position.

3.2. Two-component laser variosystems

In a two-component LVS, there are four design parameters of the scheme – the input waist position relative to the first lens, the focal lengths of the lenses, and the distance between them (Fig. 6). To form a Gaussian beam waist of constant diameter at different distances from the input waist, only two design parameters of the scheme should be changed (Table 2).

 

Fig. 6. A two-component LVS for forming a Gaussian beam waist of constant diameter at various distances from the input waist.

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Table 2. Formulas for calculating the law of change of the design parameters of a two-component LVS

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We obtain formulas for calculating the design parameters of LVSs of types A, B and C from the Table 2. Equation (13) has the following form (Fig. 6):

It is convenient to present the formulas using the optical interval $\Delta $, which is found from the solution of the quadratic equation $a{\Delta ^2} + 2b\Delta + c = 0$. The discriminant of this equation $D = {b^2} - ac$, and its solution is determined by the expression: $\Delta = \frac{{ - b \pm \sqrt D }}{a}$. Table 2 shows the formulas for calculating design parameters and coefficients a, b and c.

Thus, the dimensional calculation of the LVS scheme of any selected type is reduced to setting constant parameters and the range of variation of the leading parameter of the scheme and calculating its unknown variable parameters according to the analytical expressions given in Table 2.

The solution of the dimensional problem of a two-component LVS is possible with a positive discriminant ($D > 0$). From the expressions for $\Delta $ in the forward and reverse path of the beam through the optical system, we obtain the condition that the design parameters of any two-component laser optical system should satisfy:

For calculating the envelope diameter and the light diameter, we use the expressions:

3.3. Calculation algorithm

The calculating algorithm of the considered LVS schemes includes the following main steps.

The calculation of the LVS using the obtained relations allows cases when there is no solution or there are many solutions. In the first case, in order to obtain a solution, one should change the parameters constant for the given scheme or the restrictions imposed on it and repeat the calculation again. In the second case, we use the merit function to find the best solution. The minimum value of the merit function corresponds to the best solution.

The merit function should consider the most important parameters for the solved task. The merit function in the structural-dimensional synthesis of a LVS is:

The choice of the structure and coefficients of the merit function is made by the developer based on the conditions and requirements of the practical problem solved by the LVS. Taking into account the inverse value of the aperture value in the merit function promotes the choice of schemes with the smallest aperture ratio of lenses and simplifies carrying out the aberration synthesis of a LVS. Using the merit function makes possible to choose the most suitable structure and parameters of the LVS for solving this problem.

The practice of designing a LVS shows that typically using two-component LVSs is more preferable than using one-component LVSs.

4. Example of the structural-dimensional synthesis of laser variosystems

The LVS should form the real output waist of the Gaussian laser beam with a diameter of $2{h^{\prime}_{\textrm{w}m}} = 100.0$ µm and provide its movement $\Delta L \ge 50.0$ mm. At the LVS input, there is a Gaussian beam with parameters:

When calculating the f-number, the safety factor of the light diameter is ${K_d} = 2.6$.

Using Eq. (14), we calculate the longitudinal magnification of the LVS: ${\alpha _{G\textrm{ req}}} = {4.0^ \times }$. At the input and output of the LVS, the Rayleigh length of the Gaussian beam are: ${z_{\textrm{c1}}} = 1.54$ mm, ${z^{\prime}_{\textrm{c}m}} = 6.15$ mm.

The results of the dimensional synthesis of one- and two-component LVSs with the corresponding merit function of each solution for the mentioned initial data are presented in Table 3.

Table 3. Results of an example of dimensional synthesis of one- and two-component LVSs

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From Table 3 it follows that for this example, the best solution for structural-dimensional synthesis is the 4th solution when using a combined two-component LVS of type B, in which the distance between the input waist and the first lens with a fixed focal length and the focal length of the second lens are changed.

5. Conclusion

The features of the laws of Gaussian laser beams transformations by optical systems for structural-dimensional synthesis of a LVS are considered. Based on the laser optics theory and the analytical expressions obtained on its basis, a fairly simple technique for the analytical dimensional synthesis of one- and two-component LVSs using different combinations of variable and constant parameters to move the output waist of constant diameter has been developed.

It is important to note that the structural-dimensional synthesis stage is actually a determining stage in the development of optical systems that transforms laser radiation. Therefore, at the stage of the dimensional calculation of the LVS, the merit function was used. The proposed merit function considers the features (restrictions and requirements) of a particular practical task to the fullest extent possible and allows you to find the best solution that most simply implements the subsequent aberration synthesis of the developing LVS.

The considered LVS schemes with a combined method for changing optical characteristics can be technically implemented with a commercially available element base.