Paraxial properties of three-element zoom system for laser beam expanders based on tunable-focus lenses

Antonín Mikš; Pavel Novák

doi:10.1364/OE.23.015635

1. Introduction

Lasers are widely used in many areas of science and technology [1–8]. Since a laser beam has specific optical properties, there are required special optical systems for transformation of laser beams. Common optical systems used both in classical imaging optics and laser optics are beam expanders. In case of transformation of a homocentric light beam simple telescopic systems are being used for the beam expansion. In case of a laser beam the input Gaussian beam has to be transformed into output Gaussian beam with a different waist diameter and divergence. Thus, the design of the optical system of a laser beam expander is different because of different formulas are valid for the transformation of the Gaussian laser beam. The properties of the Gaussian beam has to be considered during the design of such laser beam expanders. In practice zoom optical systems are frequently required in various applications. For the case of classical homocentric beams the problem of an analysis and design of zoom systems is described thoroughly in literature [9–13]. However, for the case of zoom systems for transformation of Gaussian beams it is not so. Recently, we presented theoretical analysis of paraxial parameters of three element zoom systems for laser beam expanders using classical optical elements, where the change of the magnification of the system is achieved by the controlled mutual movement of individual members of the optical system [8].

In this paper we focus on the problem of a theoretical analysis of imaging properties and initial optical design of the three-element zoom optical system for laser beam expanders using lenses with a tunable focal length. Such a system makes it possible to change the transverse magnification of the Gaussian beam waist of this system without mutual movement of individual members of the system. The axial separations between individual members of such system are fixed and the change of magnification is achieved by the change of focal lengths of individual members of the system. Due to this fact the change of magnification can be very fast compared to classical systems based on mechanical movement of individual optical elements. We performed paraxial analysis of the problem and we derived equations which enable us to calculate the required optical powers of individual elements of the proposed system for laser beam expanders in dependence on the value of the axial position of the beam waist of the input Gaussian beam and the required magnification of the system. Finally, the derived equations are demonstrated on the calculation of parameters of the three-element zoom system for the laser beam expander.

The problem of influence of aberrations of optical system on the transformation of the Gaussian beam is not covered in our present work, however in case of interest one can find an analysis of this problem e.g. in [14,15].

2. Paraxial transformation of Gaussian beam by a thin lens system

Gaussian beams are paraxial solutions of the scalar Helmholtz equation and are suitable to describe the propagation of coherent laser beams. In our further analysis we will assume the most simple case of a circular Gaussian beam, which is however the most important in practice. It is a commonly known fact that the circular Gaussian beam is characterized by the function [1–8]

u = \frac{w_{0}}{w} \exp (- \frac{x^{2} + y^{2}}{w^{2}}) \exp (- i k z + i ψ - i k \frac{x^{2} + y^{2}}{2 R}),

where

R (z) = z (1 + z_{0}^{2} / z^{2}), w^{2} (z) = w_{0}^{2} (1 + z^{2} / z_{0}^{2}), ψ (z) = arc \tan (z / z_{0}), z_{0} = k w_{0}^{2} / 2,

with

k = 2 π / λ

denoting the wavenumber,

λ

denoting the wavelength of light,

2 w_{0}

denoting the narrowest diameter of a beam (the beam waist), R being the vertex radius of curvature of the beam wavefront for points located in near proximity of the z-axis of a beam, and

u = u (x, y, z)

is an arbitrary component, for example, of the electric field vector [1–8] in the orthogonal coordinate system. We characterize the divergence of a beam by the divergence angle

2 θ

that is given by the equation [1–8]

w_{0} θ = λ / π .

Figure 1 denotes the basic parameters of a Gaussian beam and its transformation by an optical system.

Fig. 1 Basic parameters of a Gaussian beam, F and F' are front and back focus of the optical system.

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Without the loss of generality we will further deal with the transformation of a Gaussian beam by the thin lens system. The equations derived for the thin lens system will be valid for the thick lens system assuming we take all the values with respect to focal points of individual optical elements or to their principal planes [16, 17]. Let us suppose an optical system consisting of a set of thin optical components (e.g. lenses) separated by air gaps. Using the formulas from ref [8], for the system of p lenses we can write (i = 1, 2, 3,…, p)

G_{i} = \frac{{f^{'}}_{i}^{2}}{q_{i}^{2} + z_{0 i}^{2}}, {q^{'}}_{i} = - q_{i} G_{i}, z_{0 i + 1} = z_{0 i} G_{i}, Δ_{i} = {f^{'}}_{i} + {f^{'}}_{i + 1} - d_{i}, q_{i + 1} = {q^{'}}_{i} + Δ_{i}

where

{f^{'}}_{i}

is the focal length of the i-th lens and

d_{i}

is the distance between i-th and i + 1-th lens. The image and object axial distance (Fig. 1) of the i-th lens can be calculated as follows

s_{i} = q_{i} - {f^{'}}_{i}, {s^{'}}_{i} = {q^{'}}_{i} + {f^{'}}_{i},

The transverse Gaussian beam waist magnification

m_{G}

of the whole system is given by [8],

m_{G}^{2} = \prod_{i = 1}^{i = p} G_{i} .

3. Three-element zoom system for laser beam expander using tunable-focus lenses

Consider a three-element optical zoom system for Gaussian beam transformation using tunable-focus lenses, which enable continuous change of the transverse Gaussian beam waist magnification $m_{G}$ of the system without the mutual movement of individual optical elements. Let the first two elements be tunable-focus lenses and the third optical element to be a classical lens with fixed focal length. The third element of the system can be easily exchanged by another lens having different focal length in order to change the parameters (magnification and image plane location) of the whole three-element system (analogously to convertors in photographic optics [18]). Such optical system allows continuous transformation of the incoming Gaussian beam. The optical scheme is given in Fig. 2. Let us further require that the image $2 {w^{'}}_{03}$ of the beam waist of the incoming Gaussian beam ( $2 w_{01}$ ) incident at the first optical element of the system is located at the image focal point of the third element ( ${s^{'}}_{3} = {f^{'}}_{3} = - s_{3}$ and $q_{3} = 0$ ) which is advantageous from the practical point of view.

Fig. 2 Three-element optical system for transformation of a Gaussian beam

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Let us now derive the equations that enable to calculate the paraxial parameters of such system i.e. the three-element optical system given in Fig. 2. For chosen separations d₁, d₂, distance L and focal length ${f^{'}}_{3}$ of the optical system our goal is to compute the values of focal lengths ${f^{'}}_{1}$ and ${f^{'}}_{2}$ of the first two tunable-focus lenses in order to achieve the desired magnification $m_{G}$ .

For our optical system it is required that the distance L between an object A (beam waist of the incoming Gaussian beam) and its image A', formed by the first two members of the optical system, does not change during the change of the magnification m_G of the system. Therefore the following relation must be valid (Fig. 2).

- s_{1} + d_{1} + {s^{'}}_{2} = L = k o n s t .

By inserting Eqs. (4) and (5) into the Eq. (7) we obtain the following equation for focal length

{f^{'}}_{1}

a_{2} {f^{'}}_{1}^{2} + a_{1} {f^{'}}_{1} + a_{0} = 0,

where

\begin{array}{l} a_{2} = (d_{1} - s_{1} - {f^{'}}_{2}) [(^{d_{1} - s_{1}) 2} - L (d_{1} - s_{1} - {f^{'}}_{2})] + z_{01}^{2} (d_{1} - s_{1} + {f^{'}}_{2} - L), \\ a_{1} = - 2 L s_{1} (^{d_{1} - {f^{'}}_{2}) 2} + 2 (d_{1} - {f^{'}}_{2}) [s_{1} (d_{1}^{2} + s_{1}^{2}) + L (s_{1}^{2} + z_{01}^{2})] \\ + z_{01}^{2} [2 d_{1} (s_{1} - d_{1}) + {f^{'}}_{2} ({f^{'}}_{2} - 2 s_{1})] - s_{1}^{2} (^{2 d_{1} - {f^{'}}_{2}) 2}, \\ a_{0} = (s_{1}^{2} + z_{01}^{2}) (d_{1} - {f^{'}}_{2}) [d_{1}^{2} - (d_{1} - {f^{'}}_{2}) (L + s_{1})] . \end{array}

In order to achieve the desired magnification

m_{G}

of the optical system it is further required that the magnification

m_{G 12} = \sqrt{G_{1} G_{2}}

between planes passing through the points A and A' satisfies the condition

m_{G 12}^{2} = \frac{{f^{'}}_{3}^{2}}{z_{01}^{2} m_{G}^{2}}

After substitution of Eq. (4) into Eq. (6), we obtain the following equation for focal length

{f^{'}}_{1}

b_{2} {f^{'}}_{1}^{2} + b_{1} {f^{'}}_{1} + b_{0} = 0

where

\begin{array}{l} b_{2} = {f^{'}}_{2}^{2} - m_{G 12}^{2} (^{{f^{'}}_{2} - d_{1} + s_{1}) 2} - m_{G 12}^{2} z_{01}^{2}, \\ b_{1} = 2 m_{G 12}^{2} (d_{1} - {f^{'}}_{2}) [s_{1} ({f^{'}}_{2} - d_{1}) + s_{1}^{2} + z_{01}^{2}], \\ b_{0} = - m_{G 12}^{2} (s_{1}^{2} + z_{01}^{2}) (^{d_{1} - {f^{'}}_{2}) 2} . \end{array}

If Eqs. (8) and (11) have a common solution, their resultant R must be equal to zero; it is valid that

R = | \begin{matrix} a_{2} & a_{1} & a_{0} & 0 \\ 0 & a_{2} & a_{1} & a_{0} \\ b_{2} & b_{1} & b_{0} & 0 \\ 0 & b_{2} & b_{1} & b_{0} \end{matrix} | .

If R = 0 should apply, then we obtain the following equation for value

{f^{'}}_{2}

; it is valid that

c_{4} {f^{'}}_{2}^{4} + c_{3} {f^{'}}_{2}^{3} + c_{2} {f^{'}}_{2}^{2} + c_{1} {f^{'}}_{2} + c_{0} = 0,

where

\begin{array}{l} c_{4} = α [z_{01}^{2} m_{G 12}^{2} (m_{G 12}^{2} - 1) - m_{G 12}^{2} s_{1}^{2} + β^{2}], \\ c_{3} = 2 d_{1} α [z_{01}^{2} m_{G 12}^{2} (1 - 2 m_{G 12}^{2}) + m_{G 12}^{2} s_{1}^{2} - 2 β^{2} + d_{1} β], \\ c_{2} = d_{1}^{2} α [z_{01}^{2} m_{G 12}^{2} (6 m_{G 12}^{2} - 1) - m_{G 12}^{2} s_{1}^{2} + 6 β^{2} - 6 d_{1} β + d_{1}^{2}], \\ c_{1} = - 2 d_{1}^{3} α [2 z_{01}^{2} m_{G 12}^{4} + 2 β^{2} - 3 d_{1} β + d_{1}^{2}], \\ c_{0} = d_{1}^{4} α [z_{01}^{2} m_{G 12}^{4} + {(β - d_{1})}^{2}], \\ α = (^{s_{1}^{2} + z_{01}^{2}) 2}, β = L + s_{1} . \end{array}

Thus all the relations needed for the computation of focal lengths

{f^{'}}_{1}

and

{f^{'}}_{2}

of the optical system are given. When calculating these focal lengths, one should follow the next steps:

1. Equation (10) is used to calculate the required magnification $m_{G 12}$ for given value of magnification $m_{G}$ .
2. Equation (15) for value ${f^{'}}_{2}$ is solved.
3. This solution is used in Eqs. (8) and (11), these are then solved for value ${f^{'}}_{1}$ and the solution common for both equations is chosen.

The issue given, i.e. calculation of focal lengths ${f^{'}}_{1}$ and ${f^{'}}_{2}$ of the first two elements (tunable-focus lenses) of the optical system of a three-element zoom optical system for transformation of Gaussian beams, is thereby solved.

4. Example

Now we will show the application of the derived equations to the calculation of paraxial parameters of the above described three-element zoom system. Assume that the beam waist radius of the input laser beam entering the optical system is $w_{01} = 0.5$ mm ( $z_{01} = 1241$ mm) and the axial distance of the beam waist from the first element of the system is $s_{1} = - 100$ mm. Wavelength of light is $λ = 0.0006328$ mm. Le us choose the design parameters as: $d_{1} = 70$ mm, $L = 140$ mm, ${f^{'}}_{3} = 100$ mm, $d_{2} = L + s_{1} - d_{1} + {f^{'}}_{3} = 70$ mm. The magnification range is $m_{G} \in 〈 1, 10 〉$ . The results of calculation using Eqs. (8)-(15) are given in Table 1, where $w_{i}$ (i = 1,2,3) stands for transverse radii of the Gaussian beam on individual elements of the zoom system, ${w^{'}}_{03}$ is the beam waist radius of the output beam emerging out from the optical system, ${s^{'}}_{3}$ is the distance of the output beam waist from the last element of our optical system and ${θ^{'}}_{3}$ is the divergence angle of the output Gaussian beam in milliradians. The linear dimensions in Table 1 are given in millimeters.

Table 1. Example of three-element zoom system for laser beam expansion based on tunable-focus lenses

View Table

5. Conclusion

A detailed theoretical analysis of paraxial properties of the three-element zoom optical system for laser beam expander using lenses with a tunable focal length was performed. The optical system of the considered three-element zoom system consists of two tunable-focus lenses and one additional classical lens which can be easily replaced by a lens of another focal length in case of need of different parameters of the output beam. The desired change of magnification is achieved without mutual movement of individual members of the optical system just by the change of optical powers of the two tunable-focus lenses. For this type of optical system we derived equations that enable to calculate the required optical powers of individual members in dependence on the value of the axial position of the beam waist of the input Gaussian beam and the required magnification of the system. These equations can be used for initial (paraxial) design of such optical system. Finally, the derived equations were applied on an example of paraxial design of a zoom system for Gaussian beam transformation.

Acknowledgments

This work has been supported by the Czech Science Foundation the grant 13-31765S.

References and links

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$s_{1} = - 100, d_{1} = 70, L = 140, d_{2} = 70, {f^{'}}_{3} = {s^{'}}_{3} = 100, w_{01} = 0.5, w_{1} = 0.5016, λ = 0.0006328$
$m_{G}$	${f^{'}}_{1}$	${f^{'}}_{2}$	w₂	w₃	${w^{'}}_{03}$	${θ^{'}}_{3}$
1.0	100.0000	−16.0822	0.1553	0.5016	0.50	0.4029
2.5	270.9496	−26.2990	0.3753	1.2501	1.25	0.1611
5	−142.9524	−34.9758	0.7500	2.5000	2.50	0.0806
7.5	−56.5638	−39.3475	1.1250	3.7500	3.75	0.0537
10.0	−35.2596	−41.9747	1.5000	5.0000	5.00	0.0403

Paraxial properties of three-element zoom system for laser beam expanders based on tunable-focus lenses

Abstract

1. Introduction

2. Paraxial transformation of Gaussian beam by a thin lens system

3. Three-element zoom system for laser beam expander using tunable-focus lenses

4. Example

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (2)

Tables (1)

Equations (15)

Optics Express