Paraxial properties of three-element zoom systems for laser beam expanders

Antonín Mikš; Pavel Novák

doi:10.1364/OE.22.021535

1. Introduction

Currently, lasers [1,2] are being widely used in many areas of science and technology. Since the laser beam has specific properties special optical systems for transformation of laser beams are required. 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 beam expansion. In case of a laser beam the input Gaussian beam has to be transformed into output Gaussian beam with different diameter and divergence. The design of the optical system of the expander is different since different formulas are valid for the transformation of a laser beam (Gaussian beam). Thus one has to consider the properties of the Gaussian beam during the design of such laser beam expander. Zoom optical systems are often required in practical applications. For the case of classical (homocentric) beams the problem of analysis and design of zoom systems is described thoroughly e.g. in Refs [3–12].

Our work is focused on the problem of theoretical analysis of paraxial properties of the three-element zoom optical system for laser beam expanders. The equations enabling the calculation of axial distances between individual elements based on the location of a beam waist of the input Gaussian beam and the desired magnification of the system are derived. Finally, an example of calculation of paraxial parameters of such a beam expander is given.

2. Basic parameters of a laser beam

Laser beam is not a homocentric light beam, it is a Gaussian beam. Different equations therefore hold for the transformation of such a beam trough the optical system with respect to classical equations that are valid for homocentric light beams [1,2,13]. In laser beams, which are very narrow, the field is concentrated along one of the axis of the chosen coordinate system (e.g. z axis). In the transverse direction the field fades relatively fast to zero whereas in the longitudinal direction it changes very slowly. Let us now restrict our analysis on the most simple case of circular Gaussian beam, which is however the most important in practice. Let $u = u (x, y, z)$ is an arbitrary component of e.g. the electric field vector in orthogonal Cartesian coordinate system, then for circular Gaussian beam one obtains [1,2,13]

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,

whereas

2 w_{0}

denotes the diameter of the beam waist (Fig. 1),

k = 2 π / λ

is the wave number and

λ

is the wavelength of light. R is vertex radius of curvature i.e. it is the radius of curvature for the points in the close proximity to z axis. The envelope of the Gaussian beam is the one-sheeted rotational hyperboloid [1,2,13].

Fig. 1 Basic parameters of a Gaussian beam

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The divergence of the beam is then characterized by the divergence angle $θ$ , which is the angle between the asymptote of the x cross section of the hyperboloid and the axis of the beam (z-axis) see Fig. 1. The following equation is valid for the Gaussian beam [1,2,13]

w_{0} θ = λ / π .

Without the loss of generality we will further deal with the paraxial transformation of a Gaussian beam by the thin lens system. The equations derived for the thin lens system are 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 [1,13–16].

As it is known from the geometrical optics theory the thin lens (Fig. 1) transforms the incident spherical wavefront with radius R to output spherical wavefront with radius R' according to the relation [1,13–16]

1 / R^{'} - 1 / R = 1 / f^{'},

where

f^{'}

is the focal length of the lens. Using Eqs. (2) and (4) we obtain the following equations for the transformation of a Gaussian beam by the thin lens system of n lenses (i = 1, 2, 3,…, n) [1,2,13]

\begin{array}{l} G_{i} = \frac{{f^{'}}_{i}^{2}}{q_{i}^{2} + z_{0 i}^{2}} = \frac{m_{i}^{2}}{1 + m_{i}^{2} {(z_{0 i} / {f^{'}}_{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}, s_{i} = q_{i} - {f^{'}}_{i}, {s^{'}}_{i} = {q^{'}}_{i} + {f^{'}}_{i}, \end{array}

where

{f^{'}}_{i}

is the focal length of the i-th lens and

d_{i}

is the distance between the i-th and the i + 1-th lens,

{s^{'}}_{i}

and

s_{i}

are image and object axial distance and

{q^{'}}_{i}

and

q_{i}

are image and object axial distance of the Gaussian beam waist from focal points (

q_{1} = s_{1} + {f^{'}}_{1}

,

z_{01} = π w_{01}^{2} / λ

),

m_{i} = {f^{'}}_{i} / q_{i}

is the transverse magnification of the lens (optical system) and

Δ_{i}

is the distance between the object-space focal point of i + 1-th lens and the image-space focal point of the i-th lens. The transverse magnification

m_{G}

of the whole system (magnification of the Gaussian beam waist after transformation of the beam by this system) is given by

m_{G}^{2} = {({w^{'}}_{0} / w_{0})}^{2} = {(θ / θ^{'})}^{2} = \prod_{i = 1}^{i = n} G_{i} .

As it can be seen from Eq. (5) function

G (q)

will have maximum for q = 0 and according to Eq. (6) the divergence

θ^{'}

of the Gaussian beam transformed by the optical system is then minimal.

3. Three-element zoom systems for laser beam expander

Figure 2 shows a principal scheme of the three-element zoom system for laser beam expander.

Fig. 2 Three-element zoom systems for laser beam expander

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The system consists of three thin lens elements with different focal lengths separated by appropriate axial distances.

By using Eqs. (5) for the three-element optical system, we obtain for the parameter $Δ_{2}$ after tedious calculation the following equation

a_{2} Δ_{2}^{2} + a_{1} Δ_{2} + a_{0} = 0,

where

\begin{array}{l} a_{2} = (^{Δ_{1} z_{01}) 2} + (^{Δ_{1} q_{1} - f_{1}^{2}) 2}, a_{1} = - 2 {f^{'}}_{2}^{2} [Δ_{1} z_{01}^{2} + q_{1} (Δ_{1} q_{1} - f_{1}^{2})], \\ a_{0} = {f^{'}}_{2}^{2} [{(z_{01} {f^{'}}_{2})}^{2} + {(q_{1} {f^{'}}_{2})}^{2} - {({f^{'}}_{1} {f^{'}}_{3} / m_{G})}^{2}] . \end{array}

We required that

q_{3} = 0

(

G_{3}

has maximum) during the derivation of Eq. (7) in order to obtain minimal divergence of the transformed Gaussian beam. For the real solution of Eq. (7) it must hold that the discriminant

D (Δ_{1}) = a_{1}^{2} - 4 a_{2} a_{0} \geq 0

. By using Eq. (8) we obtain for the parameter

Δ_{1 \min}

the following equation

b_{2} Δ_{1 \min}^{2} + b_{1} Δ_{1 \min} + b_{0} = 0,

where

b_{2} = {f^{'}}_{3}^{2} (z_{01}^{2} + q_{1}^{2}), b_{1} = - 2 q_{1} {({f^{'}}_{1} {f^{'}}_{3})}^{2}, b_{0} = {f^{'}}_{1}^{2} [{({f^{'}}_{1} {f^{'}}_{3})}^{2} - {(z_{01}^{} {f^{'}}_{2} m_{G}^{})}^{2}] .

Distance

d_{1}

then must satisfy the condition

d_{1} \geq d_{1 \min}

, where

d_{1 \min} = {f^{'}}_{1} + {f^{'}}_{2} - Δ_{1 \min}

. The axial separation d₂ of the second and the third element of the system is then given by

d_{2} = {f^{'}}_{2} + {f^{'}}_{3} - Δ_{2}

. The transverse magnification of the Gaussian beam waist

m_{G}^{}

of the three-element optical system for laser beam expander can be expressed as

m_{G}^{2} = {(\frac{{f^{'}}_{3}}{{f^{'}}_{1} {f^{'}}_{2}})}^{2} [1 + {(\frac{q_{1}}{z_{01}})}^{2}] [{(d_{1} - {f^{'}}_{1} - {f^{'}}_{2} + \frac{q_{1} {f^{'}}_{1}^{2}}{z_{01}^{2} + q_{1}^{2}})}^{2} + \frac{z_{01}^{2} {f^{'}}_{1}^{4}}{{(z_{01}^{2} + q_{1}^{2})}^{2}}] .

Due to the fact that for common lasers used in practice [14]

z_{01}

is relatively large (see Eq. (2)) the Eq. (11) for magnification

m_{G}^{}

can be approximated by the formula

{(m_{G}^{2})}_{a p r o x} \approx {[\frac{{f^{'}}_{3} ({f^{'}}_{1} + {f^{'}}_{2})}{{f^{'}}_{1} {f^{'}}_{2}} - d_{1} \frac{{f^{'}}_{3}}{{f^{'}}_{1} {f^{'}}_{2}}]}^{2} = {[\frac{{f^{'}}_{3} ({f^{'}}_{1} + {f^{'}}_{2} - d_{1})}{{f^{'}}_{1} {f^{'}}_{2}}]}^{2} = {(\frac{{f^{'}}_{3}}{{f^{'}}_{12}})}^{2},

where

{f^{'}}_{12}

is the focal length of the first two elements of the optical system. Equation (12) is very simple and it enables to determine the magnification

m_{G}

with the accuracy that is satisfactory for practice. As it can be seen from Eq. (12) magnification m_G is a linear function of distance d₁ for chosen values of focal lengths

{f^{'}}_{1}

,

{f^{'}}_{2}

and

{f^{'}}_{3}

of individual elements of given optical system. Further, one can see that Eq. (12) is equivalent to classical equation for telescopic systems for transformation of homocentric beams [3,13–16].

During the calculation of the paraxial parameters of such optical system one can also use another procedure. Firstly, one can choose focal lengths ${f^{'}}_{2}$ and ${f^{'}}_{3}$ of the second and the third element of the three-element optical system, magnification $m_{G} = m_{G \min}$ and distance d₁ = d_1min. The value of the focal length ${f^{'}}_{1 \min}$ of the first element can then be calculated using the following equation

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

where

c_{2} = {f^{'}}_{3}^{2}, c_{1} = 2 d_{1} q_{1} {f^{'}}_{3}^{2} - m_{G}^{2} z_{01}^{2} {f^{'}}_{2}^{2}, c_{0} = d_{1}^{2} {f^{'}}_{3}^{2} (q_{1}^{2} + z_{01}^{2}) ​ .

By solving Eq. (13) we have

{f^{'}}_{1 \min} = \pm \sqrt{\frac{m_{G}^{2} z_{01}^{2} {f^{'}}_{2}^{2} - 2 d_{1} q_{1} {f^{'}}_{3}^{2} \pm \sqrt{m_{G}^{2} {f^{'}}_{2}^{2} (m_{G}^{2} z_{01}^{2} {f^{'}}_{2}^{2} - 4 d_{1} q_{1} {f^{'}}_{3}^{2}) - 4 d_{1}^{2} {f^{'}}_{3}^{4}}}{2 {f^{'}}_{3}^{2}}} .

If we choose for simplicity e.g.

{f^{'}}_{2} = - {f^{'}}_{1}

, then we obtain

e_{2} {f^{'}}_{1 \min}^{4} + e_{1} {f^{'}}_{1 \min}^{2} + e_{0} = 0,

where

e_{2} = {f^{'}}_{3}^{2} - m_{G}^{2} z_{01}^{2}, e_{1} = 2 d_{1} q_{1} {f^{'}}_{3}^{2}, e_{0} = d_{1}^{2} {f^{'}}_{3}^{2} (q_{1}^{2} + z_{01}^{2}) .

By solving of Eq. (16) one obtains

{f^{'}}_{1 \min} = \pm \sqrt{\frac{d_{1} q_{1} {f^{'}}_{3}^{2} + d_{1} {f^{'}}_{3} z_{01} \sqrt{m_{G}^{2} (q_{1}^{2} + z_{01}^{2}) - {f^{'}}_{3}^{2}}}{m_{G}^{2} z_{01}^{2} - {f^{'}}_{3}^{2}}} .

By solving of Eq. (7) one can calculate the axial separation

d_{2} = {f^{'}}_{2} + {f^{'}}_{3} - Δ_{2}

. Let us now assume that the absolute value of the quantity z₀₁ is much greater than the magnitudes of other quantities in Eq. (18), which is always true in practical cases. Then the focal length

{f^{'}}_{1 \min}

can be approximately calculated as (

{f^{'}}_{2} = - {f^{'}}_{1}

)

{f^{'}}_{1 \min} \approx \pm \sqrt{d_{1} {f^{'}}_{3} / m_{G}} .

4. Example

Let us show the application of the above described analysis on an example of the three-element zoom systems for laser beam expander. If we choose ${f^{'}}_{1}^{2} = {f^{'}}_{3}$ , then according to Eq. (19) it holds that $d_{1} \approx m_{G}$ . The focal lengths of individual elements of the zoom system are chosen to be: ${f^{'}}_{1} = - 10$ mm, ${f^{'}}_{2} = 10$ mm, ${f^{'}}_{3} = 100$ mm. 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 $λ = 632.8$ nm. By solving of Eq. (9) we obtain: $d_{1 \min} = {f^{'}}_{1} + {f^{'}}_{2} - Δ_{1 \min} = 9 .9677$ mm. Thus we will choose the mutual distance between the first two elements $d_{1}$ in the range $d_{1} \in 〈 10, 50 〉$ mm.

The results of calculation using Eqs. (7) and (5) 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. The linear dimensions in Table 1 are given in millimeters.

Table 1. – Example of three-element zoom system for laser beam expansion

View Table

5. Conclusion

In our work we performed a detailed theoretical analysis of paraxial properties of the three-element zoom optical system for laser beam expanders. Equations that enable to calculate the mutual distance between individual elements of the three-element zoom optical system for laser beam expander in dependence on the value of magnification $m_{G}$ of the optical system were derived. It was shown that the magnification of such a system is approximately a linear function of the mutual distance d₁ between the first and the second element of the optical system. Example of the calculation of paraxial parameters of the three-element zoom optical system for laser beam expander was shown and the results of this calculation (for several values of magnification $m_{G}$ ) were given in Table 1. The problem of influence of aberrations of optical system on the transformation of the Gaussian beam was not studied in our work. We focused on the calculation of the paraxial parameters of the zoom system for transformation of the Gaussian beam. Those interested in the influence of aberrations on the transformed beam can find detailed information e.g. in [17]. The choice of the shape and calculation of radii of curvature of individual elements of zoom optical systems is described in detail in ref [5], thus we did not deal with this problem here.

Acknowledgments

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

References and links

1. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley-Interscience, New York, 2007).

2. H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]

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

4. K. Yamaji, Progres in Optics, Vol.VI (North-Holland Publishing Co., Amsterdam 1967).

5. A. Miks, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. 47(32), 6088–6098 (2008). [CrossRef] [PubMed]

6. T. Kryszczyński and J. Mikucki, “Structural optical design of the complex multi-group zoom systems by means of matrix optics,” Opt. Express 21(17), 19634–19647 (2013). [CrossRef] [PubMed]

7. G. Wooters and E. W. Silvertooth, “Optically Compensated Zoom Lens,” J. Opt. Soc. Am. 55(4), 347–351 (1965). [CrossRef]

8. T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012). [CrossRef]

9. S. Pal and L. Hazra, “Ab initio synthesis of linearly compensated zoom lenses by evolutionary programming,” Appl. Opt. 50(10), 1434–1441 (2011). [CrossRef] [PubMed]

10. L. Hazra and S. Pal, “A novel approach for structural synthesis of zoom systems,” Proc. SPIE 7786, 778607 (2010). [CrossRef]

11. S. Pal and L. Hazra, “Stabilization of pupils in a zoom lens with two independent movements,” Appl. Opt. 52(23), 5611–5618 (2013). [CrossRef] [PubMed]

12. S. Pal, “Aberration correction of zoom lenses using evolutionary programming,” Appl. Opt. 52(23), 5724–5732 (2013). [CrossRef] [PubMed]

13. A. Miks, Applied Optics (Czech Technical University Press, Prague, 2009).

14. http://www.edmundoptics.com

15. W. T. Welford, Aberrations of the Symmetrical Optical Systems, (Academic Press, London, 1974).

16. W. Smith, Modern optical engineering, 4th Ed., (McGraw-Hill, New York, 2007).

17. A. Miks and J. Novak, “Propagation of Gaussian beam in optical system with aberrations,” Optik: International Journal for Light and Electron Optics 114(10), 437–440 (2003). [CrossRef]

${f^{'}}_{1} = - 10, {f^{'}}_{2} = 10, {f^{'}}_{3} = 100, w_{01} = 0.5, s_{1} = - 100, w_{1} = 0.5016, λ = 0.0006328$
$m_{G}$	d₁	d₂	${s^{'}}_{3}$	${w^{'}}_{03}$	${θ^{'}}_{3} \times 10^{5}$	w₂	w₃	${(m_{G}^{})}_{a p r o x}$
10.032	10	120.006	100.000	5.016	4.017	1.004	5.016	10
20.071	20	115.002	100.000	10.036	2.008	1.505	10.036	20
30.111	30	113.334	100.000	15.055	1.338	2.007	15.055	30
40.150	40	112.500	100.000	20.075	1.004	2.509	20.075	40
50.189	50	112.000	100.000	25.095	0.803	3.011	25.095	50

Paraxial properties of three-element zoom systems for laser beam expanders

Abstract

1. Introduction

2. Basic parameters of a laser beam

3. Three-element zoom systems for laser beam expander

4. Example

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (2)

Tables (1)

Equations (19)

Optics Express