Paraxial properties of three-element zoom systems for laser beam expanders Antonín Mikš* and Pavel Novák 1
Czech Technical University in Prague, Faculty of Civil Engineering, Department of Physics, Thakurova 7, 16629 Prague, Czech Republic * [email protected]
Abstract: Our work is focused on the problem of theoretical analysis of paraxial properties of the three-element zoom optical system for laser beam expanders. Equations that enable to calculate mutual axial distances between individual elements of the system based on the axial position of the beam waist of the input Gaussian beam and the desired magnification of the system are derived. Finally, the derived equations are applied on an example of calculation of paraxial parameters of the three-element zoom system for the laser beam expander. ©2014 Optical Society of America OCIS codes: (080.2468) First-order optics; (080.2740) Geometric optical design; (220.0220) Optical design and fabrication; (140.0140) Lasers and laser optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd Ed. (Wiley-Interscience, 2007). H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. 5(10), 1550–1567 (1966). A. D. Clark, Zoom Lenses (Adam Hilger, 1973). K. Yamaji, Progres in Optics, Vol.VI (North-Holland Publishing Co., 1967). A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. 47(32), 6088–6098 (2008). 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). G. Wooters and E. W. Silvertooth, “Optically Compensated Zoom Lens,” J. Opt. Soc. Am. 55(4), 347–351 (1965). T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012). S. Pal and L. Hazra, “Ab initio synthesis of linearly compensated zoom lenses by evolutionary programming,” Appl. Opt. 50(10), 1434–1441 (2011). L. Hazra and S. Pal, “A novel approach for structural synthesis of zoom systems,” Proc. SPIE 7786, 778607 (2010). S. Pal and L. Hazra, “Stabilization of pupils in a zoom lens with two independent movements,” Appl. Opt. 52(23), 5611–5618 (2013). S. Pal, “Aberration correction of zoom lenses using evolutionary programming,” Appl. Opt. 52(23), 5724–5732 (2013). A. Mikš, Applied Optics (Czech Technical University Press, 2009). http://www.edmundoptics.com W. T. Welford, Aberrations of the Symmetrical Optical Systems, (Academic Press, 1974). W. Smith, Modern Optical Engineering, 4th Ed., (McGraw-Hill, 2007). A. Mikš 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).
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.
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21535
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=
x2 + y2 w0 x2 + y2 exp − exp ψ ikz i ik − + − , 2R w w2
(1)
where R ( z ) = z (1 + z02 / z 2 ), w2 ( z ) = w02 (1 + z 2 / z02 ), ψ ( z ) = arc tan( z / z0 ), z0 = kw02 / 2, (2)
whereas 2w0 denotes the diameter of the beam waist (Fig. 1), k = 2π / 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.
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] w0θ = λ / π .
(3)
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
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21536
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 ′,
(4)
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] fi′ 2 mi 2 = , qi′ = − qi Gi , z0i +1 = z0i Gi , 2 2 q + z0i 1 + mi ( z0i / f i ′) 2 Δ i = f i ′+ f i ′+1 − d i , qi +1 = qi′ + Δ i , si = qi − f i ′, si′ = qi′ + f i ′,
Gi =
2 i
(5)
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, si′ and si are image and object axial distance and qi′ and qi are image and object 2 / λ ), axial distance of the Gaussian beam waist from focal points ( q1 = s1 + f1′ , z01 = π w01 mi = f i ′/ qi 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 ith lens. The transverse magnification mG of the whole system (magnification of the Gaussian beam waist after transformation of the beam by this system) is given by i =n
mG2 = ( w0′ / w0 ) 2 = (θ / θ ′) 2 = ∏ Gi .
(6)
i =1
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.
The system consists of three thin lens elements with different focal lengths separated by appropriate axial distances.
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21537
By using Eqs. (5) for the three-element optical system, we obtain for the parameter Δ 2 after tedious calculation the following equation a2 Δ 22 + a1Δ 2 + a0 = 0,
(7)
where 2 a2 = ( Δ1 z01 ) 2 + ( Δ1q1 − f12 ) 2 , a1 = −2 f 2′2 Δ1 z01 + q1 ( Δ1q1 − f12 ) ,
a0 = f 2′2 ( z01 f 2′) 2 + ( q1 f 2′) 2 − ( f1′f 3′ / mG ) 2 .
(8)
We required that q3 = 0 ( G3 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 ( Δ1 ) = a12 − 4a2 a0 ≥ 0 . By using Eq. (8) we obtain for the parameter Δ1min the following equation 2 b2 Δ1min + b1Δ1min + b0 = 0,
(9)
where 2 b2 = f 3′2 ( z01 + q12 ), b1 = −2q1 ( f1′f 3′ ) 2 , b0 = f1′2 ( f1′f 3′ ) 2 − ( z01 f 2′mG ) 2 .
(10)
Distance d1 then must satisfy the condition d1 ≥ d1min , where d1min = f1′ + f 2′ − Δ1min . 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 mG of the threeelement optical system for laser beam expander can be expressed as f′ mG2 = 3 f1′ f 2′
2
2 q 2 z 2 f ′4 q f ′2 1 + 1 d1 − f1′− f 2′ + 2 1 1 2 + 2 01 1 2 2 . z01 + q1 ( z01 + q1 ) z01
(11)
Due to the fact that for common lasers used in practice [14] z01 is relatively large (see Eq. (2)) the Eq. (11) for magnification mG can be approximated by the formula 2
2
2
f ′( f ′ + f ′) f ′ f ′( f ′ + f 2′ − d1 ) f 3′ ( mG2 ) aprox ≈ 3 1 2 − d1 3 = 3 1 = f′ , ′ ′ ′ ′ f f f f1′f 2′ 1 2 1 f2 12
(12)
where f12′ 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 mG with the accuracy that is satisfactory for practice. As it can be seen from Eq. (12) magnification mG is a linear function of distance d1 for chosen values of focal lengths f1′ , 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 mG = mG min and distance d1 = ′ of the first element can then be calculated using the d1min. The value of the focal length f1min following equation ′4 + c1 f1min ′2 + c0 = 0, c2 f1min
(13)
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21538
where 2 2 c2 = f 3′2 , c1 = 2d1q1 f 3′2 − mG2 z01 f 2′2 , c0 = d12 f 3′2 ( q12 + z01 ).
(14)
By solving Eq. (13) we have ′ =± f1min
2 2 mG2 z01 f 2′2 − 2d1q1 f 3′2 ± mG2 f 2′2 ( mG2 z01 f 2′2 − 4d1q1 f 3′2 ) − 4d12 f 3′4 . (15) 2 f 3′2
If we choose for simplicity e.g. f 2′ = − f1′ , then we obtain ′4 + e1 f1min ′2 + e0 = 0, e2 f1min
(16)
2 2 e2 = f 3′2 − mG2 z01 , e1 = 2d1q1 f 3′2 , e0 = d12 f 3′2 ( q12 + z01 ).
(17)
where
By solving of Eq. (16) one obtains ′ =± f1min
2 d1q1 f 3′2 + d1 f 3′ z01 mG2 ( q12 + z01 ) − f 3′2 . 2 − f 3′2 mG2 z01
(18)
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 ′ can quantities in Eq. (18), which is always true in practical cases. Then the focal length f1min ′ ′ be approximately calculated as ( f 2 = − f1 ) ′ ≈ ± d1 f 3′ / mG . f1min
(19)
4. Example
Let us show the application of the above described analysis on an example of the threeelement zoom systems for laser beam expander. If we choose f1′2 = f 3′ , then according to Eq. (19) it holds that d1 ≈ mG . The focal lengths of individual elements of the zoom system are chosen to be: f1′ = −10 mm, f 2′ = 10 mm, f 3′ = 100 mm. The beam waist radius of the input laser beam entering the optical system is w01 = 0.5 mm ( z01 = 1241 mm) and the axial distance of the beam waist from the first element of the system is s1 = −100 mm. Wavelength of light is λ = 632.8 nm. By solving of Eq. (9) we obtain: d1min = f1′ + f 2′ − Δ1min = 9.9677 mm. Thus we will choose the mutual distance between the first two elements d1 in the range d1 ∈ 10,50 mm.
The results of calculation using Eqs. (7) and (5) are given in Table 1, where wi (i = 1,2,3) stands for transverse radii of the Gaussian beam on individual elements of the zoom system, ′ is the beam waist radius of the output beam emerging out from the optical system, s3′ is w03 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.
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21539
Table 1. – Example of three-element zoom system for laser beam expansion
f1′ = −10,
f 2′ = 10,
f 3′ = 100, w01 = 0.5, s1 = −100, w1 = 0.5016, λ = 0.0006328
mG
d1
d2
s3′
′ w03
θ3′ × 105
w2
w3
( mG ) aprox
10.032 20.071 30.111 40.150 50.189
10 20 30 40 50
120.006 115.002 113.334 112.500 112.000
100.000 100.000 100.000 100.000 100.000
5.016 10.036 15.055 20.075 25.095
4.017 2.008 1.338 1.004 0.803
1.004 1.505 2.007 2.509 3.011
5.016 10.036 15.055 20.075 25.095
10 20 30 40 50
5. Conclusion
In our work we performed a detailed theoretical analysis of paraxial properties of the threeelement 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 mG 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 mG ) 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.
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21540
Czech Technical University in Prague, Faculty of Civil Engineering, Department of Physics, Thakurova 7, 16629 Prague, Czech Republic * [email protected]
Abstract: Our work is focused on the problem of theoretical analysis of paraxial properties of the three-element zoom optical system for laser beam expanders. Equations that enable to calculate mutual axial distances between individual elements of the system based on the axial position of the beam waist of the input Gaussian beam and the desired magnification of the system are derived. Finally, the derived equations are applied on an example of calculation of paraxial parameters of the three-element zoom system for the laser beam expander. ©2014 Optical Society of America OCIS codes: (080.2468) First-order optics; (080.2740) Geometric optical design; (220.0220) Optical design and fabrication; (140.0140) Lasers and laser optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd Ed. (Wiley-Interscience, 2007). H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. 5(10), 1550–1567 (1966). A. D. Clark, Zoom Lenses (Adam Hilger, 1973). K. Yamaji, Progres in Optics, Vol.VI (North-Holland Publishing Co., 1967). A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. 47(32), 6088–6098 (2008). 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). G. Wooters and E. W. Silvertooth, “Optically Compensated Zoom Lens,” J. Opt. Soc. Am. 55(4), 347–351 (1965). T. Kryszczyński, M. Leśniewski, and J. Mikucki, “Use of matrix optics to analyze the complex multi-group zoom systems,” Proc. SPIE 8697, 86970I (2012). S. Pal and L. Hazra, “Ab initio synthesis of linearly compensated zoom lenses by evolutionary programming,” Appl. Opt. 50(10), 1434–1441 (2011). L. Hazra and S. Pal, “A novel approach for structural synthesis of zoom systems,” Proc. SPIE 7786, 778607 (2010). S. Pal and L. Hazra, “Stabilization of pupils in a zoom lens with two independent movements,” Appl. Opt. 52(23), 5611–5618 (2013). S. Pal, “Aberration correction of zoom lenses using evolutionary programming,” Appl. Opt. 52(23), 5724–5732 (2013). A. Mikš, Applied Optics (Czech Technical University Press, 2009). http://www.edmundoptics.com W. T. Welford, Aberrations of the Symmetrical Optical Systems, (Academic Press, 1974). W. Smith, Modern Optical Engineering, 4th Ed., (McGraw-Hill, 2007). A. Mikš 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).
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.
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21535
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=
x2 + y2 w0 x2 + y2 exp − exp ψ ikz i ik − + − , 2R w w2
(1)
where R ( z ) = z (1 + z02 / z 2 ), w2 ( z ) = w02 (1 + z 2 / z02 ), ψ ( z ) = arc tan( z / z0 ), z0 = kw02 / 2, (2)
whereas 2w0 denotes the diameter of the beam waist (Fig. 1), k = 2π / 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.
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] w0θ = λ / π .
(3)
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
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21536
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 ′,
(4)
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] fi′ 2 mi 2 = , qi′ = − qi Gi , z0i +1 = z0i Gi , 2 2 q + z0i 1 + mi ( z0i / f i ′) 2 Δ i = f i ′+ f i ′+1 − d i , qi +1 = qi′ + Δ i , si = qi − f i ′, si′ = qi′ + f i ′,
Gi =
2 i
(5)
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, si′ and si are image and object axial distance and qi′ and qi are image and object 2 / λ ), axial distance of the Gaussian beam waist from focal points ( q1 = s1 + f1′ , z01 = π w01 mi = f i ′/ qi 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 ith lens. The transverse magnification mG of the whole system (magnification of the Gaussian beam waist after transformation of the beam by this system) is given by i =n
mG2 = ( w0′ / w0 ) 2 = (θ / θ ′) 2 = ∏ Gi .
(6)
i =1
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.
The system consists of three thin lens elements with different focal lengths separated by appropriate axial distances.
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21537
By using Eqs. (5) for the three-element optical system, we obtain for the parameter Δ 2 after tedious calculation the following equation a2 Δ 22 + a1Δ 2 + a0 = 0,
(7)
where 2 a2 = ( Δ1 z01 ) 2 + ( Δ1q1 − f12 ) 2 , a1 = −2 f 2′2 Δ1 z01 + q1 ( Δ1q1 − f12 ) ,
a0 = f 2′2 ( z01 f 2′) 2 + ( q1 f 2′) 2 − ( f1′f 3′ / mG ) 2 .
(8)
We required that q3 = 0 ( G3 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 ( Δ1 ) = a12 − 4a2 a0 ≥ 0 . By using Eq. (8) we obtain for the parameter Δ1min the following equation 2 b2 Δ1min + b1Δ1min + b0 = 0,
(9)
where 2 b2 = f 3′2 ( z01 + q12 ), b1 = −2q1 ( f1′f 3′ ) 2 , b0 = f1′2 ( f1′f 3′ ) 2 − ( z01 f 2′mG ) 2 .
(10)
Distance d1 then must satisfy the condition d1 ≥ d1min , where d1min = f1′ + f 2′ − Δ1min . 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 mG of the threeelement optical system for laser beam expander can be expressed as f′ mG2 = 3 f1′ f 2′
2
2 q 2 z 2 f ′4 q f ′2 1 + 1 d1 − f1′− f 2′ + 2 1 1 2 + 2 01 1 2 2 . z01 + q1 ( z01 + q1 ) z01
(11)
Due to the fact that for common lasers used in practice [14] z01 is relatively large (see Eq. (2)) the Eq. (11) for magnification mG can be approximated by the formula 2
2
2
f ′( f ′ + f ′) f ′ f ′( f ′ + f 2′ − d1 ) f 3′ ( mG2 ) aprox ≈ 3 1 2 − d1 3 = 3 1 = f′ , ′ ′ ′ ′ f f f f1′f 2′ 1 2 1 f2 12
(12)
where f12′ 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 mG with the accuracy that is satisfactory for practice. As it can be seen from Eq. (12) magnification mG is a linear function of distance d1 for chosen values of focal lengths f1′ , 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 mG = mG min and distance d1 = ′ of the first element can then be calculated using the d1min. The value of the focal length f1min following equation ′4 + c1 f1min ′2 + c0 = 0, c2 f1min
(13)
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21538
where 2 2 c2 = f 3′2 , c1 = 2d1q1 f 3′2 − mG2 z01 f 2′2 , c0 = d12 f 3′2 ( q12 + z01 ).
(14)
By solving Eq. (13) we have ′ =± f1min
2 2 mG2 z01 f 2′2 − 2d1q1 f 3′2 ± mG2 f 2′2 ( mG2 z01 f 2′2 − 4d1q1 f 3′2 ) − 4d12 f 3′4 . (15) 2 f 3′2
If we choose for simplicity e.g. f 2′ = − f1′ , then we obtain ′4 + e1 f1min ′2 + e0 = 0, e2 f1min
(16)
2 2 e2 = f 3′2 − mG2 z01 , e1 = 2d1q1 f 3′2 , e0 = d12 f 3′2 ( q12 + z01 ).
(17)
where
By solving of Eq. (16) one obtains ′ =± f1min
2 d1q1 f 3′2 + d1 f 3′ z01 mG2 ( q12 + z01 ) − f 3′2 . 2 − f 3′2 mG2 z01
(18)
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 ′ can quantities in Eq. (18), which is always true in practical cases. Then the focal length f1min ′ ′ be approximately calculated as ( f 2 = − f1 ) ′ ≈ ± d1 f 3′ / mG . f1min
(19)
4. Example
Let us show the application of the above described analysis on an example of the threeelement zoom systems for laser beam expander. If we choose f1′2 = f 3′ , then according to Eq. (19) it holds that d1 ≈ mG . The focal lengths of individual elements of the zoom system are chosen to be: f1′ = −10 mm, f 2′ = 10 mm, f 3′ = 100 mm. The beam waist radius of the input laser beam entering the optical system is w01 = 0.5 mm ( z01 = 1241 mm) and the axial distance of the beam waist from the first element of the system is s1 = −100 mm. Wavelength of light is λ = 632.8 nm. By solving of Eq. (9) we obtain: d1min = f1′ + f 2′ − Δ1min = 9.9677 mm. Thus we will choose the mutual distance between the first two elements d1 in the range d1 ∈ 10,50 mm.
The results of calculation using Eqs. (7) and (5) are given in Table 1, where wi (i = 1,2,3) stands for transverse radii of the Gaussian beam on individual elements of the zoom system, ′ is the beam waist radius of the output beam emerging out from the optical system, s3′ is w03 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.
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21539
Table 1. – Example of three-element zoom system for laser beam expansion
f1′ = −10,
f 2′ = 10,
f 3′ = 100, w01 = 0.5, s1 = −100, w1 = 0.5016, λ = 0.0006328
mG
d1
d2
s3′
′ w03
θ3′ × 105
w2
w3
( mG ) aprox
10.032 20.071 30.111 40.150 50.189
10 20 30 40 50
120.006 115.002 113.334 112.500 112.000
100.000 100.000 100.000 100.000 100.000
5.016 10.036 15.055 20.075 25.095
4.017 2.008 1.338 1.004 0.803
1.004 1.505 2.007 2.509 3.011
5.016 10.036 15.055 20.075 25.095
10 20 30 40 50
5. Conclusion
In our work we performed a detailed theoretical analysis of paraxial properties of the threeelement 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 mG 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 mG ) 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.
#216326 - $15.00 USD Received 4 Jul 2014; revised 9 Aug 2014; accepted 22 Aug 2014; published 28 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021535 | OPTICS EXPRESS 21540
