Abstract

Optical variosystems that form the required diameter of the Gaussian beam waist and perform its smooth longitudinal movement with a constant diameter for laser technologies, processing materials, moving microobjects, etc., are considered. A combined method based on a movable tunable-focus lens for changing the optical characteristics of laser variosystems is proposed. The features and fundamental differences between the laws of transformation of Gaussian beams by optical systems of different structures for the dimensional synthesis of laser variosystems are discussed. Based on these laws and the theory of laser optics, a method for the dimensional synthesis of one- and two-component laser variosystems is developed, an algorithm for their automated synthesis, and an example of calculation are given.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 [14]; in technologies: laser processing of materials, additive technologies [59] 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 [1220]. 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 [2123]. 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 [2426]. 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 [3035] 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 [3034]. 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 [3034].

 figure: Fig. 1.

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]:

$$\psi ^{\prime}({x^{\prime},y^{\prime}} )= \frac{1}{{|\upbeta |}}{e^{ik({ - {s_\textrm{p}} + {{s^{\prime}_\textrm{p}}}} )}}\exp \left[ { - i\frac{k}{{2\upbeta f^{\prime}}}({{{x^{\prime 2}}} + {{y^{\prime 2}}}} )} \right]\psi \left( {\frac{{x^{\prime}}}{\upbeta },\frac{{y^{\prime}}}{\upbeta }} \right).$$
Here $k = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right.} \lambda }$ is the wave number, $\upbeta ={-} {f \mathord{\left/ {\vphantom {f {{z_\textrm{p}}}}} \right.} {{z_\textrm{p}}}} ={-} {{{{z^{\prime}_\textrm{p}}}} \mathord{\left/ {\vphantom {{{{z^{\prime}_\textrm{p}}}} {f^{\prime}}}} \right.} {f^{\prime}}} = {{{{s^{\prime}_\textrm{p}}}} \mathord{\left/ {\vphantom {{{{s^{\prime}_\textrm{p}}}} {{s_\textrm{p}}}}} \right.} {{s_\textrm{p}}}}$ is the transverse magnification in the conjugate planes P and P’, f and $f^{\prime}$ are the front and rear focal distances of the optical system, ${z_\textrm{p}}$ and ${z^{\prime}_\textrm{p}}$ are the distances that determine the position of the axial points A of the object and image $A^{\prime}$ relative to the corresponding focal planes, ${s_\textrm{p}}$ and ${s^{\prime}_\textrm{p}}$ are the distances that define the position of the axial points of the object A (image $A^{\prime}$) relative to the front (rear) main plane. The position of the conjugate planes P and P’ is determined by the formulas of Newton ${z_\textrm{p}}{z^{\prime}_\textrm{p}} = ff^{\prime}$ and Gauss $\frac{1}{{{{s^{\prime}_\textrm{p}}}}} - \frac{1}{{{s_\textrm{p}}}} = \frac{1}{{f^{\prime}}}$ [38]. Hereinafter, the characteristics and parameters of the transformed beam are denoted by a prime symbol.

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):

$$2h^{\prime} = 2h|\upbeta |,\textrm{ }\frac{1}{{{{R^{\prime}}_\Phi }}} = \frac{1}{{{\upbeta ^2}{R_\Phi }}} - \frac{1}{{\upbeta f^{\prime}}}.$$
For a transformed Gaussian beam, the envelope diameter at the level of power density ${1 \mathord{\left/ {\vphantom {1 {{e^2}}}} \right.} {{e^2}}}$ and the wavefront curvature radius have the following form [2934]:
$$2h(z )= 2{h_\textrm{w}}\sqrt {1 + {{\left( {\frac{{z - {z_\textrm{w}}}}{{{z_\textrm{c}}}}} \right)}^2}} ,\textrm{ }{R_\Phi }(z )= \frac{{{{({z - {z_\textrm{w}}} )}^2} + z_{\textrm{c} }^2}}{{z - {z_\textrm{w}}}}.$$
where z is the coordinate defining the position of the beam cross section relative to the front focal plane, ${z_\textrm{w}}$ is the distance defining the position of the waist of the transformed beam relative to the front focal plane.

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 $z^{\prime}$:

$$2h^{\prime}({z^{\prime}} )= 2{h_\textrm{w}}\left|{\frac{{z^{\prime}}}{{f^{\prime}}}} \right|\sqrt {1 + {{\left( {\frac{{{{{{f^{\prime}}^2}} \mathord{\left/ {\vphantom {{{{f^{\prime}}^2}} {z^{\prime}}}} \right.} {z^{\prime}}} + {z_\textrm{w}}}}{{{z_\textrm{c}}}}} \right)}^2}} = 2{h_\textrm{w}}\sqrt {{{\left( {\frac{{z^{\prime}}}{{f^{\prime}}}} \right)}^2} + {{\left( {\frac{{{{f^{\prime}}^2} + {z_\textrm{w}}z^{\prime}}}{{{z_\textrm{c}}f^{\prime}}}} \right)}^2}} ,$$
$$\frac{1}{{{{R^{\prime}}_\Phi }({z^{\prime}} )}} ={-} {\left( {\frac{{f^{\prime}}}{{z^{\prime}}}} \right)^2}\frac{{{{{{f^{\prime}}^2}} \mathord{\left/ {\vphantom {{{{f^{\prime}}^2}} {z^{\prime}}}} \right.} {z^{\prime}}} + {z_\textrm{w}}}}{{{{({{{{{f^{\prime}}^2}} \mathord{\left/ {\vphantom {{{{f^{\prime}}^2}} {z^{\prime}}}} \right.} {z^{\prime}}} + {z_\textrm{w}}} )}^2} + z_\textrm{c}^2}} + \frac{1}{{z^{\prime}}} = \frac{{({z_\textrm{c}^2 + z_\textrm{w}^2} )z^{\prime} + {z_\textrm{w}}{{f^{\prime}}^2}}}{{{{({{z_\textrm{w}}z^{\prime} + {{f^{\prime}}^2}} )}^2} + {{({{z_\textrm{c}}z^{\prime}} )}^2}}}.$$
Considering that the cross section of the Gaussian beam waist is a plane wavefront, from Eq. (5) we find the waist position of the transformed beam relative to the rear focal plane of the optical system:
$$z^{\prime} ={-} {z_\textrm{w}}\frac{{{{f^{\prime}}^2}}}{{z_\textrm{c}^2 + z_\textrm{w}^2}} = {z^{\prime}_\textrm{w}}.$$
Substituting Eq. (6) in Eq. (4), we find the diameter of the transformed beam waist:
$$2h^{\prime}({z^{\prime} = {{z^{\prime}_\textrm{w}}}} )= 2{h_\textrm{w}}\sqrt {\frac{{{{f^{\prime}}^2}}}{{z_\textrm{c}^2 + z_\textrm{w}^2}}} = 2{h^{\prime}_\textrm{w}}.$$
Let us analyze Eq. (5). The minimum wavefront curvature radius of the transformed beam is located in the sections ${z^{\prime}_1} ={-} ({{z_{\textrm{c} }} + {z_\textrm{w}}} )\frac{{{{f^{\prime}}^2}}}{{z_\textrm{c}^2 + z_\textrm{w}^2}}$ and ${z^{\prime}_2} = ({{z_{\textrm{c} }} - {z_\textrm{w}}} )\frac{{{{f^{\prime}}^2}}}{{z_\textrm{c}^2 + z_\textrm{w}^2}}$, it is equal to the distance between these sections and the confocal beam parameter: $|{{{R^{\prime}}_\Phi }({{{z^{\prime}}_{1,2}}} )} |= {z^{\prime}_2} - {z^{\prime}_1} = 2{z_{\textrm{c} }}\frac{{{{f^{\prime}}^2}}}{{z_\textrm{c}^2 + z_\textrm{w}^2}}$. Therefore, the Rayleigh length of the transformed beam is
$${z^{\prime}_{\textrm{c} }} = {z_{\textrm{c} }}\frac{{{{f^{\prime}}^2}}}{{z_\textrm{c}^2 + z_\textrm{w}^2}}.$$
To find the angular divergence $2\theta ^{\prime}$ of the transformed Gaussian beam, we use the fact that the cross section of the transformed beam, which is at infinity, has an optically conjugate plane in the front focal plane of the optical system (Fig. 1):
$$2\theta ^{\prime} = \frac{{2{h_F}}}{{|{f^{\prime}} |}} = \frac{{2h({z = 0} )}}{{|{f^{\prime}} |}} = \frac{{2{h_\textrm{w}}}}{{|{f^{\prime}} |}}\sqrt {1 + {{\left( {\frac{{{z_\textrm{w}}}}{{{z_\textrm{c}}}}} \right)}^2}} = \frac{{2{h_\textrm{w}}}}{{{z_\textrm{c}}}}\sqrt {\frac{{z_\textrm{c}^2 + z_\textrm{w}^2}}{{{{f^{\prime}}^2}}}} = 2\theta \sqrt {\frac{{z_\textrm{c}^2 + z_\textrm{w}^2}}{{{{f^{\prime}}^2}}}} .$$
Equations (6)–(9) obtained by the conjugate-plane method for the parameters of the transformed Gaussian beam coincide with the formulas of [3034,39].

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:

$${\upbeta _G} = \sqrt {{\alpha _G}} = \sqrt {\frac{{{{f^{\prime}}^2}}}{{z_\textrm{c}^2 + z_\textrm{w}^2}}} .$$
Thus, the spatial parameters of the Gaussian beam transformed by the optical system are determined by the following expressions:
$${z^{\prime}_\textrm{w}} ={-} {z_\textrm{w}}{\alpha _G},\textrm{ }2{h^{\prime}_\textrm{w}} = 2{h_\textrm{w}}\sqrt {{\alpha _G}} ,\textrm{ }{z^{\prime}_{\textrm{c} }} = {z_{\textrm{c} }}{\alpha _G},\textrm{ }2\theta ^{\prime} = \frac{{2\theta }}{{\sqrt {{\alpha _G}} }},\textrm{ }{\alpha _G} = \frac{{{{f^{\prime}}^2}}}{{z_\textrm{c}^2 + z_\textrm{w}^2}}.$$
When transforming Gaussian laser beams by an optical system, the following invariant holds:
$${2h_\textrm{w}}2\theta = 4{{h_\textrm{w}^2} \mathord{\left/ {\vphantom {{h_\textrm{w}^2} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = 2{h^{\prime}_\textrm{w}}2\theta ^{\prime} = 4{{h^{\prime 2}_\textrm{w}} \mathord{\left/ {\vphantom {{h^{\prime 2}_\textrm{w}} {{{z^{\prime}_\textrm{c}}}}}} \right.} {{{z^{\prime}_\textrm{c}}}}} = \textrm{const} = 4{M^2}{\lambda \mathord{\left/ {\vphantom {\lambda \pi }} \right.} \pi }.$$
From Eq. (11) it follows that it is impossible to simultaneously reduce or increase the size of the waist and the angular divergence of the Gaussian beam; as the size of the waist decreases, the Rayleigh length of the beam decreases, and vice versa.

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

$$2h^{\prime}({z^{\prime}} )= 2{h^{\prime}_\textrm{w}}\sqrt {1 + {{\left( {\frac{{z^{\prime} - {{z^{\prime}_\textrm{w}}}}}{{{{z^{\prime}_\textrm{c}}}}}} \right)}^2}} ,\textrm{ }{R^{\prime}_\Phi }({z^{\prime}} )= \frac{{{{({z^{\prime} - {{z^{\prime }}_\textrm{w}}} )^2}} + z^{\prime 2}_{c}}}{{z^{\prime} - {{z^{\prime}_\textrm{w}}}}}. $$
Equation (10) is applicable for calculating the transformation of a Gaussian beam passing through optical systems of various types: a single lens or a multicomponent optical system. For example, the parameters of a Gaussian beam after an m-component optical system are found by sequentially calculating the beam transformation by each component. A multicomponent laser optical system can be characterized by an equivalent longitudinal magnification: ${\alpha _{G\textrm{ eq }m}} = \prod\limits_{j = 1}^m {{\alpha _{G\textrm{ }j}}}$. Omitting the calculations, we provide in Table 1 the formulas for the parameters of a Gaussian beam at the output of two- and three-component laser optical systems, as well as their equivalent longitudinal magnification.

Tables Icon

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

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: ${f^{\prime}_{\min }} = {z_\textrm{c}}\sqrt {{\alpha _G}} = {{{h_\textrm{w}}} \mathord{\left/ {\vphantom {{{h_\textrm{w}}} {\theta^{\prime}}}} \right.} {\theta ^{\prime}}} = {{{{h^{\prime}_\textrm{w}}}} \mathord{\left/ {\vphantom {{{{h^{\prime}_\textrm{w}}}} \theta }} \right.} \theta } = \sqrt {{z_\textrm{c}}{{z^{\prime}_\textrm{c}}}}$.

 figure: Fig. 2.

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:

$$\frac{{2{h_{{F_m}}}}}{{|{{{f^{\prime}}_m}} |}} = \textrm{const} = \frac{{2{M^2}\lambda }}{{\pi {{h^{\prime}}_{\textrm{w}m}}}} = 2\sqrt {\frac{{{M^2}\lambda }}{{\pi {{z^{\prime}}_{{\textrm{c} }m}}}}} .$$
This expression allows for the selected LVS scheme to obtain the law of variation of its design parameters, which provides a change of the longitudinal position of the output waist of constant diameter. In this case, the longitudinal magnification of the LVS remains constant:
$${\alpha _{G\textrm{ eq }m}} = {({{{2{{h^{\prime}}_{\textrm{w}m}}} \mathord{\left/ {\vphantom {{2{{h^{\prime}}_{\textrm{w}m}}} {2{h_{\textrm{w1}}}}}} \right.} {2{h_{\textrm{w1}}}}}} )^2} = {{{{z^{\prime}}_{\textrm{c}m}}} \mathord{\left/ {\vphantom {{{{z^{\prime}}_{\textrm{c}m}}} {{z_{\textrm{c}1}}}}} \right.} {{z_{\textrm{c}1}}}} = {({{{2{\theta_1}} \mathord{\left/ {\vphantom {{2{\theta_1}} {2{{\theta^{\prime}}_m}}}} \right.} {2{{\theta^{\prime}}_m}}}} )^2} = \textrm{const} = {\alpha _{G\textrm{ req}}}.$$
Here ${\alpha _{G\textrm{ req}}}$ is the required longitudinal magnification of LVS.

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 $H^{\prime}$:

$${s_\textrm{w}} = {z_\textrm{w}} - f^{\prime},\textrm{ }{s^{\prime}_\textrm{w}} = {z^{\prime}_\textrm{w}} + f^{\prime}.$$
An important step in the calculation of an LVS is structural-dimensional synthesis, which includes determining the required number of components and dimensional parameters of the scheme in “thin” components, at which a Gaussian laser beam with the required spatial parameters is formed.

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:

$$\frac{{2{h_F}}}{{|{f^{\prime}} |}} = \frac{{2{h_\textrm{w}}}}{{|{f^{\prime}} |}}\sqrt {1 + {{\left( {\frac{{{s_\textrm{w}} + f^{\prime}}}{{{z_\textrm{c}}}}} \right)}^2}} = \frac{{2{M^2}\lambda }}{{\pi {{h^{\prime}_\textrm{w}}}}},\,\textrm{or}\,\frac{{\sqrt {z_{\textrm{c} }^2 + {{({{s_\textrm{w}} + f^{\prime}} )}^2}} }}{{|{f^{\prime}} |}} = \frac{{{h_\textrm{w}}}}{{{{h^{\prime}_\textrm{w}}}}},$$
From this, considering Eq. (14), we obtain the dependence of the lens focal length on the distance between the lens and the input beam waist:
$$f^{\prime}({{s_\textrm{w}}} )= \frac{{ - {s_\textrm{w}}{\alpha _{G\textrm{ req}}} \pm {z_\textrm{c}}\sqrt {[{1 - {\alpha_{G\textrm{ req}}} + {{({{{{s_\textrm{w}}} \mathord{\left/ {\vphantom {{{s_\textrm{w}}} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}}} )}^2}} ]{\alpha _{G\textrm{ req}}}} }}{{{\alpha _{G\textrm{ req}}} - 1}}.$$

 figure: Fig. 3.

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:

$$L({{s_\textrm{w}}} )={-} {s_\textrm{w}} \mp {z_\textrm{c}}\sqrt {[{1 - {\alpha_{G\textrm{ req}}} + {{({{{{s_\textrm{w}}} \mathord{\left/ {\vphantom {{{s_\textrm{w}}} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}}} )}^2}} ]{\alpha _{G\textrm{ req}}}} .$$
For a particular case ${\alpha _{G\textrm{ req}}} = 1$, the dependences of the required focal length and the length of the variosystem are determined by the expressions $f^{\prime}({{s_\textrm{w}}} )={-} \frac{{s_\textrm{w}^2 + z_\textrm{c}^2}}{{2{s_\textrm{w}}}}$, $L({{s_\textrm{w}}} )={-} 2{s_\textrm{w}}$.

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.

 figure: Fig. 4.

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.

 figure: Fig. 5.

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 ${{\Delta {s_\textrm{w}}} \mathord{\left/ {\vphantom {{\Delta {s_\textrm{w}}} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = \Delta {\bar{s}_\textrm{w}}$).

Download Full Size | PPT Slide | PDF

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:

$$2{h_L}({{s_\textrm{w}}} )= 2{h_\textrm{w}}\sqrt {1 + {{\left( {\frac{{{s_\textrm{w}}}}{{{z_\textrm{c}}}}} \right)}^2}} = 2{h^{\prime}_\textrm{w}}\sqrt {1 + {{\left( {\frac{{L + {s_\textrm{w}}}}{{{{z^{\prime}_\textrm{c}}}}}} \right)}^2}} ,\textrm{ }2{h_{\textrm{ca}}}({{s_\textrm{w}}} )= 2{K_d}{h_L}({{s_\textrm{w}}} ),$$
where ${K_d}$ is the safety factor (excess ratio) of the lens light diameter relative to the diameter of the laser beam [34,35,40].

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).

 figure: Fig. 6.

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

Download Full Size | PPT Slide | PDF

Tables Icon

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

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):

$$\frac{{2{h_{{F_2}}}}}{{|{{{f^{\prime}}_2}} |}} = 2{h_{\textrm{w1}}}\left|{\frac{\Delta }{{{{f^{\prime}}_1}{{f^{\prime}}_2}}}} \right|\sqrt {1 + {{\left( {\frac{{{{f^{\prime}}_1} + {s_{\textrm{w1}}} + {{f^{\prime ^2}_1} \mathord{\left/ {\vphantom {{f^{\prime 2}_1} \Delta }} \right.} \Delta }}}{{{z_{\textrm{c1}}}}}} \right)}^2}} = \frac{{2{M^2}\lambda }}{{\pi {{h^{\prime}}_{\textrm{w2}}}}},$$
$$\textrm{or}\,\frac{{\sqrt {{{({{z_{\textrm{c1}}}\Delta } )}^2} + {{[{({{{f^{\prime}}_1} + {s_{\textrm{w1}}}} )\Delta + f^{\prime ^2}_1} ]}^2}} }}{{|{{{f^{\prime}}_1}{{f^{\prime}}_2}} |}} = \frac{{{h_{\textrm{w1}}}}}{{{{h^{\prime}}_{\textrm{w2}}}}}.$$
Here, to find the diameter of the beam in the front focal plane of the second lens $2{h_{{F_2}}}$, using the reverse ray trace through the first lens we found the section of the beam conjugated to the plane ${F_2}$.

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:

$${\left( {\frac{{{{f^{\prime}}_1}}}{{{{f^{\prime}}_2}}}} \right)^2}\left[ {1 + {{\left( {\frac{{{{f^{\prime}}_2} - {{s^{\prime}}_{\textrm{w2}}}}}{{{{z^{\prime}}_{\textrm{c}2}}}}} \right)}^2}} \right] \ge \frac{1}{{{\alpha _{G\textrm{ req}}}}} \ge \frac{{{{({{{{{f^{\prime}}_1}} \mathord{\left/ {\vphantom {{{{f^{\prime}}_1}} {{{f^{\prime}}_2}}}} \right.} {{{f^{\prime}}_2}}}} )}^2}}}{{1 + {{[{{{({{{f^{\prime}}_1} + {s_{\textrm{w1}}}} )} \mathord{\left/ {\vphantom {{({{{f^{\prime}}_1} + {s_{\textrm{w1}}}} )} {{z_{\textrm{c}1}}}}} \right.} {{z_{\textrm{c}1}}}}} ]}^2}}}.$$
When calculating a two-component LVS, the lens movement can be set as follows: the first lens ${s_{\textrm{w1}}}({{\delta_{L1}}} )= s_{\textrm{w1}}^{(0 )} - {\delta _{L1}}$; the second lens $d({{\delta_{L2}}} )= {d^{(0 )}} + {\delta _{L2}}$. Here $s_{\textrm{w1}}^{(0 )}$ and ${s_{\textrm{w1}}}$ are the initial and current position of the input beam waist relative to the first lens, ${d^{(0 )}}$ and d are the initial and current distances between the lenses; ${\delta _{Lj}}$ is the lens movement: ${\delta _{Lj}} < 0$ – to the left relative to the initial position, ${\delta _{Lj}} > 0$ – to the right relative to the initial position.

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

$$\begin{array}{l} 2{h_{L1}} = 2{h_{\textrm{w1}}}\sqrt {1 + {{\left( {\frac{{{s_{\textrm{w1}}}}}{{{z_{\textrm{c1}}}}}} \right)}^2}} ,\textrm{ }2{h_{1\textrm{ ca}}} = 2{K_d}{h_{L1}},\\ \textrm{2}{h_{L2}} = 2{h_{\textrm{w1}}}\sqrt {{{\left( {1 - \frac{d}{{{{f^{\prime}}_1}}}} \right)}^2} + {{\left[ {\frac{{d - ({1 - {d \mathord{\left/ {\vphantom {d {{{f^{\prime}}_1}}}} \right.} {{{f^{\prime}}_1}}}} ){s_{\textrm{w1}}}}}{{{z_{\textrm{c1}}}}}} \right]}^2}} = 2{{h^{\prime}}_{\textrm{w2}}}\sqrt {1 + {{\left( {\frac{{L + {s_{\textrm{w1}}} - d}}{{{{z^{\prime}}_{\textrm{c2}}}}}} \right)}^2}} ,\textrm{ }2{h_{2\textrm{ ca}}} = 2{K_d}{h_{L2}}. \end{array}$$

3.3. Calculation algorithm

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

  • 1. Setting the initial data: laser Gaussian beam parameters at the optical system input, the required diameter of the output waist and the range of its movement, constant parameters of the LVS scheme of this type, all important restrictions on the selected scheme and LVS parameters.
  • 2. The calculation of the required longitudinal magnification of the LVS using Eq. (14).
  • 3. Determining the possible range of the leading parameter variation of the scheme and calculating two variable parameters of the scheme according to Eq. (17) and Eq. (18) or according to the formulas shown in Table 2. Checking the validity of all parameters of the scheme and the law of their change with the given restrictions. Finding all possible solutions of the problem of dimensional synthesis and calculating the merit function of each solution.
  • 4. The choice of the best solution according to the value of the merit function.
In the behavior of the structural-dimensional synthesis of a LVS, it is necessary to set the constraints and the merit function. The restrictions contain all allowable ranges for its longitudinal and transverse dimensions, the range of variation of the focal length of the lenses, the minimum allowable f-number of all lenses in the system, and other parameters. F-number characterizes the aperture ratio of the lens, which plays a large role in the next stage of LVS development - aberration synthesis.

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:

$$Fz = \sum\limits_{k = 1}^N {{p_k}{m_k}{t_k}} ,$$
where ${t_k}$ is the LVS parameter; ${p_k}$ and ${m_k}$ are the weight and scale factors of this parameter.

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:

  • • radiation wavelength $\lambda = 1.064$ µm;
  • • beam quality factor ${M^2} = 1.2$;
  • • waist diameter $2{h_{\textrm{w1}}} = 50.0$ µm.
Restrictions in the structural-dimensional synthesis of a LVS:
  • I. Single-component LVS
    • 1. $- {s_\textrm{w}} = 45.0 \ldots 100.0$ mm – range of allowable distances between an input waist and the first lens;
    • 2. $N{d_{\min }} = 5.0$ – minimum f-number;
    • 3. ${f^{\prime}_{\min }} = 15.0$ mm – minimum realizable focal length of a variable focal length lens.
  • II. Two-component LVS
    • 1. $- {s_{\textrm{w1}}} = 10.0 \ldots 100.0$mm – range of allowable distances between an input waist and the first lens;
    • 2. $d = 50.0 \ldots 400.0$ mm – allowable distance between lenses;
    • 3. ${s^{\prime}_{\textrm{w2}}} = 0 \ldots 200.0$ mm – range of allowable distances between the second lens and the output waist;
    • 4. $N{d_{\min }} = 5.0$ – minimum f-number of both LVS components;
    • 5. ${f^{\prime}_{\min }} = 15.0$ mm – minimum realizable focal length of a variable focal length lens of a LVS scheme.
In the calculation, we used an merit function considering 3 parameters: $\Delta d$ is the value of the lens displacement, $\Delta f$ is the range of variation of the focal length of the tunable lens, $N{d_{\min }}$ is the minimum f-number of the LVS lenses, $Fz = {p_1}\left|{\frac{{\Delta d}}{{{m_1}}}} \right|+ {p_2}\left|{\frac{{{m_2}}}{{\Delta f}}} \right|+ {p_3}\frac{{{m_3}}}{{N{d_{\min }}}}$. The following weights are accepted here: ${p_1} = 2.5$, ${p_2} = 1.5$, ${p_3} = 1.0$. The scale factors for a one- and two-component LVS of types A and B are ${m_1} = 25.0$ mm, ${m_2} = 35.0$ mm, ${m_3} = 5.0$, and for a two-component LVS of type C ${m_1} = 25.0$ mm, ${m_2} = 5.0$ mm, ${m_3} = 5.0$.

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.

Tables Icon

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

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.

Funding

Russian Foundation for Basic Research (18-38-20155).

Disclosures

The authors declare no conflicts of interest.

References

1. H. Masumoto, “Development of zoom lenses for camera and technical topics: design examples, analysis, optical design method, aspherical lenses, and manufacturing,” Proc. SPIE 3482, 202 (1998). [CrossRef]  

2. S. C. Park and K. Kim, “Video camera zoom lens design using lens modules,” Proc. SPIE 2539, 192 (1995). [CrossRef]  

3. K. Matsusaka, S. Ozawa, R. Yoshida, T. Yuasa, and Y. Souma, “Ultracompact optical zoom lens for mobile phone,” Proc. SPIE 6502, 650203 (2007). [CrossRef]  

4. D. Doering, T. Milde, and M. Hanft, “Zoom lens design for projection optics,” Proc. SPIE 9626, 962617 (2015). [CrossRef]  

5. M. Baumann, V. Krause, G. Bergweiler, M. Flaischerowitz, and J. Banik, “Local heat treatment of high strength steels with zoom-optics and 10kW-diode laser,” Proc. SPIE 8239, 82390J (2012). [CrossRef]  

6. C. E. Webb and J. D. C. Jones, eds., Handbook of Laser Technology and Applications, Volume II. Laser Design and Laser Systems (IoP, 2004).

7. S. Katayama, Handbook of Laser Welding Technologies (Woodhead, 2013).

8. A. G. Grigoryants, I. N. Shiganov, and A. I. Misyurov, Technological Processes of Laser Treatment (BMSTU, 2006).

9. E. D. Vax, M. N. Milenky, and L. G. Saprykin, Practice of Precision Laser Processing (Technosphere, 2013).

10. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]  

11. Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019). [CrossRef]  

12. G. Wooters and E. W. Silvertooth, “Optically compensated zoom lens,” J. Opt. Soc. Am. 55(4), 347–351 (1965). [CrossRef]  

13. A. D. Clark, Zoom Lenses (Adam Hilger, 1973).

14. I. I. Pakhomov, Zoom Systems (Mashinostroenie, 1976).

15. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type,” Appl. Opt. 21(12), 2174–2183 (1982). [CrossRef]  

16. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 2: Generalization of Yamaji Type V,” Appl. Opt. 21(22), 4045–4053 (1982). [CrossRef]  

17. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 3: Five-component type,” Appl. Opt. 22(4), 541–553 (1983). [CrossRef]  

18. T. Kryszczynski, “Paraxial determination of the general four-component zoom system with mechanical compensation,” Proc. SPIE 2539, 180–191 (1995). [CrossRef]  

19. A. M. Khorokhorov, D. E. Piskunov, and A. F. Shirankov, “First-order method of zoom lens design by means of generalized parameters,” J. Opt. Soc. Am. A 33(8), 1537–1545 (2016). [CrossRef]  

20. O. V. Rozhkov, D. E. Piskunov, P. A. Nosov, V. Y. Pavlov, A. M. Khorokhorov, and A. F. Shirankov, “Bauman MSTU scientific school «Zoom lens design»: features of theory and practice,” Comput. Opt. 42(1), 72–83 (2018). [CrossRef]  

21. S. Szapiel and C. Greenhalgh, “Freeform based lateral-shift compact zoom system: theory and computer simulations,” Proc. SPIE 10627, 6 (2018). [CrossRef]  

22. L. W. Alvarez, “Two-Element Variable-Power Spherical Lens,” Patent US 3305294 (1967).

23. J. Babington, “Alvarez lens systems: theory and applications,” Proc. SPIE 9626, 962615 (2015). [CrossRef]  

24. A. Mikš and J. Novák, “Analysis of two-element zoom systems based on variable power lenses,” Opt. Express 18(7), 6797–6810 (2010). [CrossRef]  

25. A. Mikš and J. Novák, “Three-component double conjugate zoom lens system from tunable focus lenses,” Appl. Opt. 52(4), 862–865 (2013). [CrossRef]  

26. S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007). [CrossRef]  

27. www.optotune.com

28. www.corning.com

29. F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006). [CrossRef]  

30. S. William and C. Chang, Principles of Lasers and Optics (Cambridge University Press, 2005).

31. H. Kogelnik and T. Li, “Laser beams and resonator,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef]  

32. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1974).

33. I. I. Pakhomov, O. V. Rozhkov, and V. N. Rozhdestvin, Optoelectronic Quantum Devices (Radio i Svyaz’, 1982).

34. I. I. Pakhomov and A. B. Tsibulya, “Computational methods for laser optical system design,” J. Russ. Laser Res. 9(3), 321–430 (1988). [CrossRef]  

35. P. A. Nosov, V. Y. Pavlov, I. I. Pakhomov, and A. F. Shirankov, “Aberrational synthesis of optical systems intended for the conversion of laser beams,” J. Opt. Technol. 78(9), 586–593 (2011). [CrossRef]  

36. ISO 11146-1 “Lasers and laser-related equipment. Test methods for laser beam widths, divergence angles and propagation ratios – Part 1: Stigmatic and simple astigmatic beams,” 2005.

37. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

38. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1990).

39. A. Mikš and P. Novák, “Paraxial properties of three-element zoom systems for laser beam expanders,” Opt. Express 22(18), 21535–21540 (2014). [CrossRef]  

40. P. A. Nosov, D. E. Piskunov, and A. F. Shirankov, “Laser varifocal system synthesis for longitudinal Gaussian beam shifting,” Appl. Opt. 58(13), 3347–3353 (2019). [CrossRef]  

References

  • View by:

  1. H. Masumoto, “Development of zoom lenses for camera and technical topics: design examples, analysis, optical design method, aspherical lenses, and manufacturing,” Proc. SPIE 3482, 202 (1998).
    [Crossref]
  2. S. C. Park and K. Kim, “Video camera zoom lens design using lens modules,” Proc. SPIE 2539, 192 (1995).
    [Crossref]
  3. K. Matsusaka, S. Ozawa, R. Yoshida, T. Yuasa, and Y. Souma, “Ultracompact optical zoom lens for mobile phone,” Proc. SPIE 6502, 650203 (2007).
    [Crossref]
  4. D. Doering, T. Milde, and M. Hanft, “Zoom lens design for projection optics,” Proc. SPIE 9626, 962617 (2015).
    [Crossref]
  5. M. Baumann, V. Krause, G. Bergweiler, M. Flaischerowitz, and J. Banik, “Local heat treatment of high strength steels with zoom-optics and 10kW-diode laser,” Proc. SPIE 8239, 82390J (2012).
    [Crossref]
  6. C. E. Webb and J. D. C. Jones, eds., Handbook of Laser Technology and Applications, Volume II. Laser Design and Laser Systems (IoP, 2004).
  7. S. Katayama, Handbook of Laser Welding Technologies (Woodhead, 2013).
  8. A. G. Grigoryants, I. N. Shiganov, and A. I. Misyurov, Technological Processes of Laser Treatment (BMSTU, 2006).
  9. E. D. Vax, M. N. Milenky, and L. G. Saprykin, Practice of Precision Laser Processing (Technosphere, 2013).
  10. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [Crossref]
  11. Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019).
    [Crossref]
  12. G. Wooters and E. W. Silvertooth, “Optically compensated zoom lens,” J. Opt. Soc. Am. 55(4), 347–351 (1965).
    [Crossref]
  13. A. D. Clark, Zoom Lenses (Adam Hilger, 1973).
  14. I. I. Pakhomov, Zoom Systems (Mashinostroenie, 1976).
  15. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type,” Appl. Opt. 21(12), 2174–2183 (1982).
    [Crossref]
  16. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 2: Generalization of Yamaji Type V,” Appl. Opt. 21(22), 4045–4053 (1982).
    [Crossref]
  17. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 3: Five-component type,” Appl. Opt. 22(4), 541–553 (1983).
    [Crossref]
  18. T. Kryszczynski, “Paraxial determination of the general four-component zoom system with mechanical compensation,” Proc. SPIE 2539, 180–191 (1995).
    [Crossref]
  19. A. M. Khorokhorov, D. E. Piskunov, and A. F. Shirankov, “First-order method of zoom lens design by means of generalized parameters,” J. Opt. Soc. Am. A 33(8), 1537–1545 (2016).
    [Crossref]
  20. O. V. Rozhkov, D. E. Piskunov, P. A. Nosov, V. Y. Pavlov, A. M. Khorokhorov, and A. F. Shirankov, “Bauman MSTU scientific school «Zoom lens design»: features of theory and practice,” Comput. Opt. 42(1), 72–83 (2018).
    [Crossref]
  21. S. Szapiel and C. Greenhalgh, “Freeform based lateral-shift compact zoom system: theory and computer simulations,” Proc. SPIE 10627, 6 (2018).
    [Crossref]
  22. L. W. Alvarez, “Two-Element Variable-Power Spherical Lens,” Patent US 3305294 (1967).
  23. J. Babington, “Alvarez lens systems: theory and applications,” Proc. SPIE 9626, 962615 (2015).
    [Crossref]
  24. A. Mikš and J. Novák, “Analysis of two-element zoom systems based on variable power lenses,” Opt. Express 18(7), 6797–6810 (2010).
    [Crossref]
  25. A. Mikš and J. Novák, “Three-component double conjugate zoom lens system from tunable focus lenses,” Appl. Opt. 52(4), 862–865 (2013).
    [Crossref]
  26. S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
    [Crossref]
  27. www.optotune.com
  28. www.corning.com
  29. F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
    [Crossref]
  30. S. William and C. Chang, Principles of Lasers and Optics (Cambridge University Press, 2005).
  31. H. Kogelnik and T. Li, “Laser beams and resonator,” Appl. Opt. 5(10), 1550–1567 (1966).
    [Crossref]
  32. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1974).
  33. I. I. Pakhomov, O. V. Rozhkov, and V. N. Rozhdestvin, Optoelectronic Quantum Devices (Radio i Svyaz’, 1982).
  34. I. I. Pakhomov and A. B. Tsibulya, “Computational methods for laser optical system design,” J. Russ. Laser Res. 9(3), 321–430 (1988).
    [Crossref]
  35. P. A. Nosov, V. Y. Pavlov, I. I. Pakhomov, and A. F. Shirankov, “Aberrational synthesis of optical systems intended for the conversion of laser beams,” J. Opt. Technol. 78(9), 586–593 (2011).
    [Crossref]
  36. ISO 11146-1 “Lasers and laser-related equipment. Test methods for laser beam widths, divergence angles and propagation ratios – Part 1: Stigmatic and simple astigmatic beams,” 2005.
  37. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  38. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1990).
  39. A. Mikš and P. Novák, “Paraxial properties of three-element zoom systems for laser beam expanders,” Opt. Express 22(18), 21535–21540 (2014).
    [Crossref]
  40. P. A. Nosov, D. E. Piskunov, and A. F. Shirankov, “Laser varifocal system synthesis for longitudinal Gaussian beam shifting,” Appl. Opt. 58(13), 3347–3353 (2019).
    [Crossref]

2019 (2)

Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019).
[Crossref]

P. A. Nosov, D. E. Piskunov, and A. F. Shirankov, “Laser varifocal system synthesis for longitudinal Gaussian beam shifting,” Appl. Opt. 58(13), 3347–3353 (2019).
[Crossref]

2018 (2)

O. V. Rozhkov, D. E. Piskunov, P. A. Nosov, V. Y. Pavlov, A. M. Khorokhorov, and A. F. Shirankov, “Bauman MSTU scientific school «Zoom lens design»: features of theory and practice,” Comput. Opt. 42(1), 72–83 (2018).
[Crossref]

S. Szapiel and C. Greenhalgh, “Freeform based lateral-shift compact zoom system: theory and computer simulations,” Proc. SPIE 10627, 6 (2018).
[Crossref]

2016 (1)

2015 (2)

J. Babington, “Alvarez lens systems: theory and applications,” Proc. SPIE 9626, 962615 (2015).
[Crossref]

D. Doering, T. Milde, and M. Hanft, “Zoom lens design for projection optics,” Proc. SPIE 9626, 962617 (2015).
[Crossref]

2014 (1)

2013 (1)

2012 (1)

M. Baumann, V. Krause, G. Bergweiler, M. Flaischerowitz, and J. Banik, “Local heat treatment of high strength steels with zoom-optics and 10kW-diode laser,” Proc. SPIE 8239, 82390J (2012).
[Crossref]

2011 (1)

2010 (1)

2007 (2)

K. Matsusaka, S. Ozawa, R. Yoshida, T. Yuasa, and Y. Souma, “Ultracompact optical zoom lens for mobile phone,” Proc. SPIE 6502, 650203 (2007).
[Crossref]

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[Crossref]

2006 (1)

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[Crossref]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

1998 (1)

H. Masumoto, “Development of zoom lenses for camera and technical topics: design examples, analysis, optical design method, aspherical lenses, and manufacturing,” Proc. SPIE 3482, 202 (1998).
[Crossref]

1995 (2)

S. C. Park and K. Kim, “Video camera zoom lens design using lens modules,” Proc. SPIE 2539, 192 (1995).
[Crossref]

T. Kryszczynski, “Paraxial determination of the general four-component zoom system with mechanical compensation,” Proc. SPIE 2539, 180–191 (1995).
[Crossref]

1988 (1)

I. I. Pakhomov and A. B. Tsibulya, “Computational methods for laser optical system design,” J. Russ. Laser Res. 9(3), 321–430 (1988).
[Crossref]

1983 (1)

1982 (2)

1966 (1)

1965 (1)

Alvarez, L. W.

L. W. Alvarez, “Two-Element Variable-Power Spherical Lens,” Patent US 3305294 (1967).

Babington, J.

J. Babington, “Alvarez lens systems: theory and applications,” Proc. SPIE 9626, 962615 (2015).
[Crossref]

Banik, J.

M. Baumann, V. Krause, G. Bergweiler, M. Flaischerowitz, and J. Banik, “Local heat treatment of high strength steels with zoom-optics and 10kW-diode laser,” Proc. SPIE 8239, 82390J (2012).
[Crossref]

Batshev, V. I.

Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019).
[Crossref]

Baumann, M.

M. Baumann, V. Krause, G. Bergweiler, M. Flaischerowitz, and J. Banik, “Local heat treatment of high strength steels with zoom-optics and 10kW-diode laser,” Proc. SPIE 8239, 82390J (2012).
[Crossref]

Berge, B.

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[Crossref]

Bergweiler, G.

M. Baumann, V. Krause, G. Bergweiler, M. Flaischerowitz, and J. Banik, “Local heat treatment of high strength steels with zoom-optics and 10kW-diode laser,” Proc. SPIE 8239, 82390J (2012).
[Crossref]

Bräuer, A.

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[Crossref]

Chang, C.

S. William and C. Chang, Principles of Lasers and Optics (Cambridge University Press, 2005).

Clark, A. D.

A. D. Clark, Zoom Lenses (Adam Hilger, 1973).

Deladi, S.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[Crossref]

Doering, D.

D. Doering, T. Milde, and M. Hanft, “Zoom lens design for projection optics,” Proc. SPIE 9626, 962617 (2015).
[Crossref]

Flaischerowitz, M.

M. Baumann, V. Krause, G. Bergweiler, M. Flaischerowitz, and J. Banik, “Local heat treatment of high strength steels with zoom-optics and 10kW-diode laser,” Proc. SPIE 8239, 82390J (2012).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Greenhalgh, C.

S. Szapiel and C. Greenhalgh, “Freeform based lateral-shift compact zoom system: theory and computer simulations,” Proc. SPIE 10627, 6 (2018).
[Crossref]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

Grigoryants, A. G.

A. G. Grigoryants, I. N. Shiganov, and A. I. Misyurov, Technological Processes of Laser Treatment (BMSTU, 2006).

Hanft, M.

D. Doering, T. Milde, and M. Hanft, “Zoom lens design for projection optics,” Proc. SPIE 9626, 962617 (2015).
[Crossref]

Helwegen, I.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[Crossref]

Hendriks, B. H. W.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[Crossref]

Katayama, S.

S. Katayama, Handbook of Laser Welding Technologies (Woodhead, 2013).

Khorokhorov, A. M.

O. V. Rozhkov, D. E. Piskunov, P. A. Nosov, V. Y. Pavlov, A. M. Khorokhorov, and A. F. Shirankov, “Bauman MSTU scientific school «Zoom lens design»: features of theory and practice,” Comput. Opt. 42(1), 72–83 (2018).
[Crossref]

A. M. Khorokhorov, D. E. Piskunov, and A. F. Shirankov, “First-order method of zoom lens design by means of generalized parameters,” J. Opt. Soc. Am. A 33(8), 1537–1545 (2016).
[Crossref]

Kim, K.

S. C. Park and K. Kim, “Video camera zoom lens design using lens modules,” Proc. SPIE 2539, 192 (1995).
[Crossref]

Kogelnik, H.

Kozlov, A. B.

Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019).
[Crossref]

Krasin, G. K.

Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019).
[Crossref]

Krause, V.

M. Baumann, V. Krause, G. Bergweiler, M. Flaischerowitz, and J. Banik, “Local heat treatment of high strength steels with zoom-optics and 10kW-diode laser,” Proc. SPIE 8239, 82390J (2012).
[Crossref]

Kryszczynski, T.

T. Kryszczynski, “Paraxial determination of the general four-component zoom system with mechanical compensation,” Proc. SPIE 2539, 180–191 (1995).
[Crossref]

Kuiper, S.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[Crossref]

Li, T.

Machikhin, A. S.

Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019).
[Crossref]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1974).

Masumoto, H.

H. Masumoto, “Development of zoom lenses for camera and technical topics: design examples, analysis, optical design method, aspherical lenses, and manufacturing,” Proc. SPIE 3482, 202 (1998).
[Crossref]

Matsusaka, K.

K. Matsusaka, S. Ozawa, R. Yoshida, T. Yuasa, and Y. Souma, “Ultracompact optical zoom lens for mobile phone,” Proc. SPIE 6502, 650203 (2007).
[Crossref]

Mikš, A.

Milde, T.

D. Doering, T. Milde, and M. Hanft, “Zoom lens design for projection optics,” Proc. SPIE 9626, 962617 (2015).
[Crossref]

Milenky, M. N.

E. D. Vax, M. N. Milenky, and L. G. Saprykin, Practice of Precision Laser Processing (Technosphere, 2013).

Misyurov, A. I.

A. G. Grigoryants, I. N. Shiganov, and A. I. Misyurov, Technological Processes of Laser Treatment (BMSTU, 2006).

Nosov, P. A.

Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019).
[Crossref]

P. A. Nosov, D. E. Piskunov, and A. F. Shirankov, “Laser varifocal system synthesis for longitudinal Gaussian beam shifting,” Appl. Opt. 58(13), 3347–3353 (2019).
[Crossref]

O. V. Rozhkov, D. E. Piskunov, P. A. Nosov, V. Y. Pavlov, A. M. Khorokhorov, and A. F. Shirankov, “Bauman MSTU scientific school «Zoom lens design»: features of theory and practice,” Comput. Opt. 42(1), 72–83 (2018).
[Crossref]

P. A. Nosov, V. Y. Pavlov, I. I. Pakhomov, and A. F. Shirankov, “Aberrational synthesis of optical systems intended for the conversion of laser beams,” J. Opt. Technol. 78(9), 586–593 (2011).
[Crossref]

Novák, J.

Novák, P.

Ozawa, S.

K. Matsusaka, S. Ozawa, R. Yoshida, T. Yuasa, and Y. Souma, “Ultracompact optical zoom lens for mobile phone,” Proc. SPIE 6502, 650203 (2007).
[Crossref]

Pakhomov, I. I.

P. A. Nosov, V. Y. Pavlov, I. I. Pakhomov, and A. F. Shirankov, “Aberrational synthesis of optical systems intended for the conversion of laser beams,” J. Opt. Technol. 78(9), 586–593 (2011).
[Crossref]

I. I. Pakhomov and A. B. Tsibulya, “Computational methods for laser optical system design,” J. Russ. Laser Res. 9(3), 321–430 (1988).
[Crossref]

I. I. Pakhomov, Zoom Systems (Mashinostroenie, 1976).

I. I. Pakhomov, O. V. Rozhkov, and V. N. Rozhdestvin, Optoelectronic Quantum Devices (Radio i Svyaz’, 1982).

Park, S. C.

S. C. Park and K. Kim, “Video camera zoom lens design using lens modules,” Proc. SPIE 2539, 192 (1995).
[Crossref]

Pavlov, V. Y.

O. V. Rozhkov, D. E. Piskunov, P. A. Nosov, V. Y. Pavlov, A. M. Khorokhorov, and A. F. Shirankov, “Bauman MSTU scientific school «Zoom lens design»: features of theory and practice,” Comput. Opt. 42(1), 72–83 (2018).
[Crossref]

P. A. Nosov, V. Y. Pavlov, I. I. Pakhomov, and A. F. Shirankov, “Aberrational synthesis of optical systems intended for the conversion of laser beams,” J. Opt. Technol. 78(9), 586–593 (2011).
[Crossref]

Pichugina, Y. V.

Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019).
[Crossref]

Piskunov, D. E.

Rozhdestvin, V. N.

I. I. Pakhomov, O. V. Rozhkov, and V. N. Rozhdestvin, Optoelectronic Quantum Devices (Radio i Svyaz’, 1982).

Rozhkov, O. V.

O. V. Rozhkov, D. E. Piskunov, P. A. Nosov, V. Y. Pavlov, A. M. Khorokhorov, and A. F. Shirankov, “Bauman MSTU scientific school «Zoom lens design»: features of theory and practice,” Comput. Opt. 42(1), 72–83 (2018).
[Crossref]

I. I. Pakhomov, O. V. Rozhkov, and V. N. Rozhdestvin, Optoelectronic Quantum Devices (Radio i Svyaz’, 1982).

Saprykin, L. G.

E. D. Vax, M. N. Milenky, and L. G. Saprykin, Practice of Precision Laser Processing (Technosphere, 2013).

Schreiber, P.

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[Crossref]

Shiganov, I. N.

A. G. Grigoryants, I. N. Shiganov, and A. I. Misyurov, Technological Processes of Laser Treatment (BMSTU, 2006).

Shirankov, A. F.

Silvertooth, E. W.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1990).

Souma, Y.

K. Matsusaka, S. Ozawa, R. Yoshida, T. Yuasa, and Y. Souma, “Ultracompact optical zoom lens for mobile phone,” Proc. SPIE 6502, 650203 (2007).
[Crossref]

Suijver, J. F.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[Crossref]

Szapiel, S.

S. Szapiel and C. Greenhalgh, “Freeform based lateral-shift compact zoom system: theory and computer simulations,” Proc. SPIE 10627, 6 (2018).
[Crossref]

Tanaka, K.

Tsibulya, A. B.

I. I. Pakhomov and A. B. Tsibulya, “Computational methods for laser optical system design,” J. Russ. Laser Res. 9(3), 321–430 (1988).
[Crossref]

Vax, E. D.

E. D. Vax, M. N. Milenky, and L. G. Saprykin, Practice of Precision Laser Processing (Technosphere, 2013).

William, S.

S. William and C. Chang, Principles of Lasers and Optics (Cambridge University Press, 2005).

Wippermann, F. C.

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[Crossref]

Wooters, G.

Yoshida, R.

K. Matsusaka, S. Ozawa, R. Yoshida, T. Yuasa, and Y. Souma, “Ultracompact optical zoom lens for mobile phone,” Proc. SPIE 6502, 650203 (2007).
[Crossref]

Yuasa, T.

K. Matsusaka, S. Ozawa, R. Yoshida, T. Yuasa, and Y. Souma, “Ultracompact optical zoom lens for mobile phone,” Proc. SPIE 6502, 650203 (2007).
[Crossref]

Appl. Opt. (6)

Comput. Opt. (1)

O. V. Rozhkov, D. E. Piskunov, P. A. Nosov, V. Y. Pavlov, A. M. Khorokhorov, and A. F. Shirankov, “Bauman MSTU scientific school «Zoom lens design»: features of theory and practice,” Comput. Opt. 42(1), 72–83 (2018).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Technol. (1)

J. Russ. Laser Res. (1)

I. I. Pakhomov and A. B. Tsibulya, “Computational methods for laser optical system design,” J. Russ. Laser Res. 9(3), 321–430 (1988).
[Crossref]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

Opt. Express (2)

Proc. SPIE (11)

Y. V. Pichugina, P. A. Nosov, V. I. Batshev, A. S. Machikhin, A. B. Kozlov, and G. K. Krasin, “3D manipulation of micro-objects based on optical tweezers using acousto-optic deflector and variofocal system,” Proc. SPIE 11060, 46 (2019).
[Crossref]

T. Kryszczynski, “Paraxial determination of the general four-component zoom system with mechanical compensation,” Proc. SPIE 2539, 180–191 (1995).
[Crossref]

H. Masumoto, “Development of zoom lenses for camera and technical topics: design examples, analysis, optical design method, aspherical lenses, and manufacturing,” Proc. SPIE 3482, 202 (1998).
[Crossref]

S. C. Park and K. Kim, “Video camera zoom lens design using lens modules,” Proc. SPIE 2539, 192 (1995).
[Crossref]

K. Matsusaka, S. Ozawa, R. Yoshida, T. Yuasa, and Y. Souma, “Ultracompact optical zoom lens for mobile phone,” Proc. SPIE 6502, 650203 (2007).
[Crossref]

D. Doering, T. Milde, and M. Hanft, “Zoom lens design for projection optics,” Proc. SPIE 9626, 962617 (2015).
[Crossref]

M. Baumann, V. Krause, G. Bergweiler, M. Flaischerowitz, and J. Banik, “Local heat treatment of high strength steels with zoom-optics and 10kW-diode laser,” Proc. SPIE 8239, 82390J (2012).
[Crossref]

F. C. Wippermann, P. Schreiber, A. Bräuer, and B. Berge, “Mechanically assisted liquid lens zoom system for mobile phone cameras,” Proc. SPIE 6289, 62890T (2006).
[Crossref]

J. Babington, “Alvarez lens systems: theory and applications,” Proc. SPIE 9626, 962615 (2015).
[Crossref]

S. Szapiel and C. Greenhalgh, “Freeform based lateral-shift compact zoom system: theory and computer simulations,” Proc. SPIE 10627, 6 (2018).
[Crossref]

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[Crossref]

Other (15)

www.optotune.com

www.corning.com

L. W. Alvarez, “Two-Element Variable-Power Spherical Lens,” Patent US 3305294 (1967).

S. William and C. Chang, Principles of Lasers and Optics (Cambridge University Press, 2005).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, 1974).

I. I. Pakhomov, O. V. Rozhkov, and V. N. Rozhdestvin, Optoelectronic Quantum Devices (Radio i Svyaz’, 1982).

ISO 11146-1 “Lasers and laser-related equipment. Test methods for laser beam widths, divergence angles and propagation ratios – Part 1: Stigmatic and simple astigmatic beams,” 2005.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1990).

C. E. Webb and J. D. C. Jones, eds., Handbook of Laser Technology and Applications, Volume II. Laser Design and Laser Systems (IoP, 2004).

S. Katayama, Handbook of Laser Welding Technologies (Woodhead, 2013).

A. G. Grigoryants, I. N. Shiganov, and A. I. Misyurov, Technological Processes of Laser Treatment (BMSTU, 2006).

E. D. Vax, M. N. Milenky, and L. G. Saprykin, Practice of Precision Laser Processing (Technosphere, 2013).

A. D. Clark, Zoom Lenses (Adam Hilger, 1973).

I. I. Pakhomov, Zoom Systems (Mashinostroenie, 1976).

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Figures (6)

Fig. 1.
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.
Fig. 2.
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.
Fig. 3.
Fig. 3. A single-component LVS for the formation of a Gaussian beam waist of constant diameter at various distances from the input waist.
Fig. 4.
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.
Fig. 5.
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 ${{\Delta {s_\textrm{w}}} \mathord{\left/ {\vphantom {{\Delta {s_\textrm{w}}} {{z_\textrm{c}}}}} \right.} {{z_\textrm{c}}}} = \Delta {\bar{s}_\textrm{w}}$).
Fig. 6.
Fig. 6. A two-component LVS for forming a Gaussian beam waist of constant diameter at various distances from the input waist.

Tables (3)

Tables Icon

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

Tables Icon

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

Tables Icon

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

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

ψ ( x , y ) = 1 | β | e i k ( s p + s p ) exp [ i k 2 β f ( x 2 + y 2 ) ] ψ ( x β , y β ) .
2 h = 2 h | β | ,   1 R Φ = 1 β 2 R Φ 1 β f .
2 h ( z ) = 2 h w 1 + ( z z w z c ) 2 ,   R Φ ( z ) = ( z z w ) 2 + z c 2 z z w .
2 h ( z ) = 2 h w | z f | 1 + ( f 2 / f 2 z z + z w z c ) 2 = 2 h w ( z f ) 2 + ( f 2 + z w z z c f ) 2 ,
1 R Φ ( z ) = ( f z ) 2 f 2 / f 2 z z + z w ( f 2 / f 2 z z + z w ) 2 + z c 2 + 1 z = ( z c 2 + z w 2 ) z + z w f 2 ( z w z + f 2 ) 2 + ( z c z ) 2 .
z = z w f 2 z c 2 + z w 2 = z w .
2 h ( z = z w ) = 2 h w f 2 z c 2 + z w 2 = 2 h w .
z c = z c f 2 z c 2 + z w 2 .
2 θ = 2 h F | f | = 2 h ( z = 0 ) | f | = 2 h w | f | 1 + ( z w z c ) 2 = 2 h w z c z c 2 + z w 2 f 2 = 2 θ z c 2 + z w 2 f 2 .
β G = α G = f 2 z c 2 + z w 2 .
z w = z w α G ,   2 h w = 2 h w α G ,   z c = z c α G ,   2 θ = 2 θ α G ,   α G = f 2 z c 2 + z w 2 .
2 h w 2 θ = 4 h w 2 / h w 2 z c z c = 2 h w 2 θ = 4 h w 2 / h w 2 z c z c = const = 4 M 2 λ / λ π π .
2 h ( z ) = 2 h w 1 + ( z z w z c ) 2 ,   R Φ ( z ) = ( z z w ) 2 + z c 2 z z w .
2 h F m | f m | = const = 2 M 2 λ π h w m = 2 M 2 λ π z c m .
α G  eq  m = ( 2 h w m / 2 h w m 2 h w1 2 h w1 ) 2 = z c m / z c m z c 1 z c 1 = ( 2 θ 1 / 2 θ 1 2 θ m 2 θ m ) 2 = const = α G  req .
s w = z w f ,   s w = z w + f .
2 h F | f | = 2 h w | f | 1 + ( s w + f z c ) 2 = 2 M 2 λ π h w , or z c 2 + ( s w + f ) 2 | f | = h w h w ,
f ( s w ) = s w α G  req ± z c [ 1 α G  req + ( s w / s w z c z c ) 2 ] α G  req α G  req 1 .
L ( s w ) = s w z c [ 1 α G  req + ( s w / s w z c z c ) 2 ] α G  req .
2 h L ( s w ) = 2 h w 1 + ( s w z c ) 2 = 2 h w 1 + ( L + s w z c ) 2 ,   2 h ca ( s w ) = 2 K d h L ( s w ) ,
2 h F 2 | f 2 | = 2 h w1 | Δ f 1 f 2 | 1 + ( f 1 + s w1 + f 1 2 / f 1 2 Δ Δ z c1 ) 2 = 2 M 2 λ π h w2 ,
or ( z c1 Δ ) 2 + [ ( f 1 + s w1 ) Δ + f 1 2 ] 2 | f 1 f 2 | = h w1 h w2 .
( f 1 f 2 ) 2 [ 1 + ( f 2 s w2 z c 2 ) 2 ] 1 α G  req ( f 1 / f 1 f 2 f 2 ) 2 1 + [ ( f 1 + s w1 ) / ( f 1 + s w1 ) z c 1 z c 1 ] 2 .
2 h L 1 = 2 h w1 1 + ( s w1 z c1 ) 2 ,   2 h 1  ca = 2 K d h L 1 , 2 h L 2 = 2 h w1 ( 1 d f 1 ) 2 + [ d ( 1 d / d f 1 f 1 ) s w1 z c1 ] 2 = 2 h w2 1 + ( L + s w1 d z c2 ) 2 ,   2 h 2  ca = 2 K d h L 2 .
F z = k = 1 N p k m k t k ,

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