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Paraxial properties of two-element zoom systems for Gaussian beam transformation
Optik 126 (2015) 4249–4253
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Paraxial properties of two-element zoom systems for Gaussian beam transformation Antonín Mikˇs, Pavel Novák ∗ Czech Technical University in Prague, Faculty of Civil Engineering, Department of Physics, Thakurova 7, 16629 Prague, Czech Republic
a r t i c l e
i n f o
Article history: Received 12 August 2014 Accepted 23 August 2015 Keywords: Zoom systems Geometric optics First-order optical design Gaussian beams
a b s t r a c t This work deals with a problem of paraxial analysis of optical systems for transformation of Gaussian beams. General equations describing the paraxial parameters of a two-element zoom system for transformation of a Gaussian beam are derived. It is shown that such a zoom lens has different kinematics with respect to classical zoom system designed for homocentric beams, which depends on the parameters of the transformed Gaussian beam. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Lasers [1–3] are currently used in many areas of science and technology. Different devices and instruments that take advantage of the properties of a laser beam are being developed nowadays. The laser beam used in these devices needs to be transformed using a special optical system to achieve the required properties and parameters of the output beam e.g., divergence, position of the beam waist etc. In many application areas (e.g., laser scanning, cutting, annealing, welding etc.) a continuous change of the laser beam parameters is required. This can be achieved by employing an optical system that makes possible a continuous change of some of its parameters (e.g., focal length and transverse magnification), which is usually referred to as a zoom lens. The problem of analysis and design of zoom lenses for classical (homocentric) light beams can be found e.g., in Refs. [4–21]. However, as it is well-known the laser beam is not homocentric, it is a Gaussian beam, therefore different equations hold for the transformation of such a beam through the optical system with respect to classical equations that are valid for homocentric light beams [1–3,22–28]. Thus one cannot use the well-known equations for the design of classical zoom lenses for homocentric light beams [4–22] to design a zoom system for transformation of a Gaussian beam. Different equations have to
∗ Corresponding author. Tel.: +420 224357919. E-mail address: [email protected] (P. Novák). http://dx.doi.org/10.1016/j.ijleo.2015.08.123 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
be derived that enable the paraxial design of the zoom lens for the transformation of Gaussian beam. In case the optical system made from classical optical elements makes possible to change the focal length in a continuous manner it has to have at least two members separated by a specific distance. By changing the mutual positions of those two elements one can achieve a continuous change of focal length of this system in a given range. The aim of our work is to derive equations for calculation of paraxial properties of the two-element zoom system for transforming a Gaussian beam. 2. Basic parameters of a Gaussian beam While the theory of Gaussian beams is well-known, we briefly outline it here for the benefit of the reader. Assume a harmonic electromagnetic wave field propagating through the homogenous isotropic medium. Such field is then described by the Helmholtz equation [1,2,22].
∇ 2 u(x, y, z) + k2 u(x, y, z) = 0, where, ∇ 2
(1)
is a Laplace operator, k = 2/ is the wave number, is the wavelength of light and u = u(x, y, z) is an arbitrary component of e.g., the electric field vector E [1,2,22] in orthogonal Cartesian coordinate system. Eq. (1) describes the electromagnetic field in the whole space. 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.
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A. Mikˇs, P. Novák / Optik 126 (2015) 4249–4253
In such a case the solution of Eq. (1) is sought in the following form [1,2,22]. (2) u(x, y, z) = ϕ(x, y, z) exp(−ikz), √ where, i = −1. By substitution of Eq. (2) into Eq. (1) we obtain 2
2
∂ ϕ ∂ ϕ ∂ϕ + − 2ik = 0, ∂x2 ∂ y2 ∂z
(3)
where we neglected the term ∂2 ϕ/∂z2 due to the reasons given above. Eq. (3) is a parabolic equation [1,2,22] for function ϕ(x, y, z). Let us now restrict our analysis on the most simple case of circular Gaussian beam, which is however the most important in practice. By substitution of solution of Eq. (3) into Eq. (2) one obtains [1,2,22–28] w0 exp w
u=
−
x 2 + y2 w2
exp
−ikz + i
− ik
x 2 + y2 2R
,
(4)
where, R = R(z) = z(1 + z02 /z 2 ) w2 = w2 (z) = w02 (1 + z 2 /z02 ) =
(z) = arctan(z/z0 )
,
(5)
z0 = kw02 /2 whereas, 2w0 denotes the diameter of the beam waist. As it is obvious from Eq. (4) the amplitude of the beam decrease in transverse direction with a Gaussian function, which is the reason for the denotation as Gaussian beam. In case that R1 and R2 are the radii of curvature of the mirrors in laser resonator and p is their mutual distance, it holds z02 =
p(R1 − p)(R2 − p)(R1 + R2 − p) (R1 + R2 − 2p)2
.
(6)
Equation for the wavefront of such Gaussian beam then according to Eq. (4) takes the form z+
x 2 + y2 − = z1 = konst. 2R k
(7)
Due to the fact that the wavelength emitted by e.g., He – Ne laser is very small (He–Ne = 0.0006328 mm) and the beam waist radius w0 of common He – Ne lasers is usually 0.3 – 0.5 mm [29], for z0 ≈ 1000 mm and z = 1000 mm the third term on the left hand side of the Eq. (7) is /k ≈ 0.00008, which can be neglected with respect to remaining two terms. Equation of the wavefront then can be written as x 2 + y2 = z1 − z. 2R
Fig. 1. Basic parameters of a Gaussian beam.
which is the equation of the one-sheeted rotational hyperboloid. The divergence of the beam is then characterized by the divergence angle , which is the angle between the asymptote of the x cross section of the hyperboloid and the axis of the beam (z-axis). The total angular spread of the beam is then given by angle 2. From Eq. (10) we obtain (tan ≈ ) w0 = /.
Fig. 1 shows schematically the basic parameters of the Gaussian beam and its transformation by optical system.
3. Transformation of a Gaussian beam by an optical system 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 will be valid for the thick lens system assuming we take all the values with respect to focal points F and F of individual optical elements (Fig. 2) or to their principal planes [22,30–32]. As it is known from the geometrical optics theory the thin lens (Fig. 2) transforms the incident spherical wavefront with radius R to output spherical wavefront with radius R according to the relation [22,30–32] 1/R − 1/R = 1/f ,
(8)
Eq. (8) is the equation of a rotational surface, whereas, R is its vertex radius of curvature i.e., it is the radius of curvature for the points in the close proximity to z-axis. From Eq. (5) we can find the minimum value of the radius of curvature Re by employing a condition for the extremum (dR/dz = 0) Re = 2z0 .
(9)
One can see that the radius of curvature changes from R =∞ for z = 0 to minimum value of R = Re for z = z0 and then it again increases to infinity for z→ ∞. The shape of the Gaussian beam in space is determined by the change of the amplitude. It is usual to describe the shape of the Gaussian beam by the surface (envelope) where the amplitude decrease to 1/e (e = 2.71828 is the Euler number) of the value on the beam axis. From Eq. (4) we have x2 + y2 = w02 (1 + z 2 /z02 ),
(10)
(11)
Fig. 2. Transformation of a Gaussian beam by the optical system.
(12)
A. Mikˇs, P. Novák / Optik 126 (2015) 4249–4253
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Fig. 3. Transformation of a Gaussian beam by two consecutive elements of a thin lens system. Fig. 4. Two element optical system (F and F are focal points and P and P are principal points of the system).
where, f is the focal length of the lens. Using the previous equation and Eq. (5) we obtain (Fig. 2) f
G=
2
q2 + (Re /2)
f
=
2
q
w0 =
2
q2 + z02
=
1 + m2 (z0 /f )
2
= −qG
Re = Re G
The transverse Gaussian beam waist magnification mG of the whole system is given by
m2
Re /k, w0 =
m2G =
.
Re /k
The meaning of individual symbols is evident from Fig. 2. The quantity mG is the Gaussian beam waist magnification (magnification of the beam waist) and m = f /q is the transverse magnification of the lens (optical system). Using Eq. (13) we obtain qq = −f
2
+ m2G (Re /2) = −f
2
For the case of the two-element optical system (Fig. 4), assuming that we know the focal lengths f1 , f2 and the distance d between the two lenses, we obtain the following relations using the equations from the previous section 2
q1 =
+ m2G z02
2
f − (Re /2) m2G
q = (1/mG ) q = −mG
2
2
.
(14)
Gi =
,
=
sF =
2
Eq. (14) is the generalization of the Newton’s lens formula for the Gaussian beam. For Re = z0 = 0 Eq. (14) becomes a standard Newton’s lens formula for homocentric beam. Let us now assume we have a thin lens system in air. Fig. 3 shows the situation for 2 consecutive lenses (ith and i + 1th lens). Using Eq. (13) then for the thin lens system of n lenses (i = 1, 2, 3,. . ., n) one obtains
Re,i+1 = Rei Gi
f 1 ±
2
q2
f − (Re /2) m2G
2 f i 2 2 qi + (Rei /2) qi = −qi Gi
(17)
4. Transformation of a Gaussian beam by the two-element zoom system
s = q − f , s = q + f
2
Gi .
i=1
(13)
w0 = w0 = / √ mG = w0 /w0 = G
i=n
(15)
i = fi + fi+1 − di
qi+1 = qi + i
m2G =
2
(f1 −
2
(f1 f2 /mG ) − (Re1 /2)
2
2 f 2 q1 (f 1 − q1 ) − Re1 /4 2
2
(f 1 − q1 ) + (Re1 /2)
−f1 (1
−
f1 /),
2 (f1 f 2 ) = 2 2 q1 ) + (Re1 /2) q = q1 − f1 − sF ,
sF
=
f2 (1 f
−
q2 + (Re1 /2)
q =
q2
+
f2
2
,
f2 /)
2 2
−
=
(18)
m2 1 + m2 (Re1 /2f )
2
sF
q = −qm2G , = f1 + f2 − d, f = f1 f2 /
where, f is the focal length of the two-element optical system and m = f /q is the transverse magnification of the system. The meaning of individual symbols if evident from Fig. 4. Now we will deal with the paraxial design of the zoom lens for Gaussian beams. Such system should hold fixed the distance L between the object A (beam waist of the incoming Gaussian beam) and the image A (beam waist of the output Gaussian beam) during the change of the magnification mG of the system (change of the diameter of the output beam waist). In case of such requirement, it holds (Fig. 4) −q − sF + d + sF + q = L = konst.
fi
2
(19)
Where, is the focal length of the ith lens and di is the distance between the ith and the i + 1th lens. In case we are interested in image and object axial distances we can calculate them as
By substitution of Eq. (18) into Eq. (19) we obtain for the parameter after tedious calculation the following equation
si = qi − fi , si = qi + fi .
4 + a3 3 + a2 2 + a1 + a0 = 0,
(16)
(20)
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A. Mikˇs, P. Novák / Optik 126 (2015) 4249–4253 Table 1 Two-element zoom system for Gaussian beam (solution 1).
where, 2
2
˛ = L − 2(f1 + f2 ), ˇ = f1 + f2 , = f1 f2 , ı = Re1 /2 a3 = 2˛
2
a2 = ˛2 + 2ˇ + ı(m2G + 1) a1 = 2˛ˇ
a0 = ˇ2 − (m2G + 1)/mG
f1 = 30, f2 = 25 w01 = 0.3, L = 20
.
(21)
2
For chosen focal lengths f1 and f2 of both members of the optical system, Gaussian beam waist magnification mG of the optical system, distance L between the object and the image (distance between the beam waists of the input and output beams), radius of Re1 /2z and the wavethe input Gaussian beam waist w01 = length of the input light beam entering the optical system, one can calculate the quantity by solving Eq. (20). The mutual distance d between the two lenses forming the zoom system can be calculated using d = f1 + f2 − .
(22)
By substitution of the calculated value into Eq. (18) one can obtain q1 and q2 . If we choose the magnification of the Gaussian
beam waist in an interval mG ∈ mG min , mG max then we obtain the mutual distance d = d(mG ) by the above described procedure using Eqs. (18), (20), (21) and (22). The axial distances s1 and s2 can then be determined by substitution of calculated q1 and q2 into relations (Fig. 4) s1 = q1 − f1 , s2 = q2 + f2 .
(23)
The transverse diameter of the Gaussian beam (i.e., the diameter given by the amplitude decrease to 1/e of the axial value) on individual members (lenses) of given two-element optical system can be calculated using Eq. (1). Thus the given problem of determination of paraxial parameters of the two-element zoom system is solved. In case of image axial distance s2 from the second lens of the twoelement system is negative, additional optical system with focal length f3 ≥ (−s1 + d)max has to be used. The position of the addi of the tional system is given by the requirement that the image w02 beam waist created by our two-element zoom system is placed in the object focal point of the additional optical system. The image of created by this additional optical system then the beam waist w02 would be located in the image focal point of this system. Re1 /2 of the In case we set the beam waist diameter w01 = incident Gaussian beam equal to zero i.e., we assume homocentric beam Eq. (20) then converts to the classical equation for a twoelement zoom system as it will be shown further. Eq. (20) can be modified to the form
( + ˛) + ˇ
2
− (m2G + 1)/mG
2
1 − (ımG )
2
= 0.
(24)
By setting Re1 = 0 i.e., ı = Re1 /2 = 0 in Eq. (24) (homocentric beam), we obtain 2 + ˛ + ˇ + (m2 + 1)/m = 0,
(25)
whereas, m = f /q (see Eq. (18)) is the transverse magnification of the optical system. Equation (25) then gives us the classical solution of the two-element zoom lens system. As it can be seen from Eqs. (20) and (25) zoom systems for the transformation of Gaussian beams have different kinematics of individual components with respect to zoom systems designed for classical homocentric beams.
mG 0.2 0.3 0.4 0.5 0.6 0.7 0.8
s1 −20.4782 −49.4240 −74.9422 −93.6256 −100.5899 −86.1914 −24.0868
D 46.8088 49.8700 51.5312 52.6068 53.3674 53.9343 54.3851
s2 −47.2869 −79.2940 −106.4733 −126.2324 −133.9574 −120.1257 −58.4719
Table 2 Two-element zoom system for Gaussian beam (solution 2). f1 = 30, f2 = 25, w01 = 0.3, L = 20 mG 0.2 0.3 0.4 0.5 0.6 0.7 0.8
s1 144.8605 180.7550 216.8562 250.9363 280.1627 298.6764 287.7910
d 61.9360 59.0084 57.4810 56.5407 55.9096 55.4608 55.1137
s2 102.9244 141.7466 179.3752 214.3957 244.2531 263.2155 252.6773
5. Examples of calculation of the two-element zoom system for Gaussian beam transformation In order to give the reader a rough idea of the differences in kinematics of individual components of the two-element zoom system for Gaussian and classical (homocentric) beams we will present the following example. Table 1 shows the results of the calculation of paraxial parameters of the two-element zoom lens for Gaussian beam transformation. As a laser we choose e.g., He – Ne laser HNL020R from Thorlabs [29] with beam waist radius w0 = 0.3 mm. Individual members of the optical system of the zoom lens are assumed to be thin lenses for simplicity. Parameters of the zoom system were chosen as f1 = 30 mm, f2 = 25 mm and L = 20 mm. By solving Eq. (20) and using Eqs. (18) and (23) we obtain for the first real root of the Eq. (20) the values given in Table 1. Table 2 then represents the results obtained for the second real root of Eq. (20). Table 3 then shows values calculated for homocentric light beam w01 = 0 for comparison. Values of length quantities in Tables 1–3 are given in millimeters. As it can be seen from Table 1 the axial distance of the image from the second element of the zoom lens s2 is negative therefore we will use additional optical system placed behind our zoom lens to image the beam waist. The focal length of the additional system can be chosen e.g., f3 = 160 mm and it is located in the distance f3 + L = 180 mm from the object plane of the zoom system (i.e., from the beam waist of the beam incident at the zoom lens). Image of the beam waist of the Gaussian beam will then be located of w03 at the image focal point F3 of this additional system i.e., at distance f3 = 160 mm from the additional system. As it can be seen further from Tables 2 and 3 the solution of the zoom for the Gaussian beam Table 3 Classical two-element zoom system for homocentric light beam. f1 = 30, f2 = 25, w01 = 0, L = 20 m −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8
s1 103.5990 105.7188 115.0806 130.6868 153.8083 187.8883 240.4160
d 76.3325 66.7891 61.7204 58.7340 56.9042 55.7868 55.1387
s2 47.2665 58.9297 73.3602 91.9528 116.9042 152.1015 205.2774
A. Mikˇs, P. Novák / Optik 126 (2015) 4249–4253
transformation differs significantly from the solution of the classical two-element zoom system for imaging by the homocentric light beam. 6. Conclusion In our work a detailed analysis of the paraxial transformation of the Gaussian beam by an optical system was performed. The equations for calculation of paraxial parameters of the two-element zoom system enabling continuous magnification of the beam waist diameter were derived. It was shown that the mutual distance of the two elements forming the two-element zoom system can be found by solving Eq. (20), which is a fourth order equation. It was shown that the zoom systems for Gaussian beam transformation have different kinematics of the motion of individual elements of zoom lens than the classical zoom systems for homocentric beams. Further it was shown that for the case of homocentric beam (Re1 = 0) the derived Eq. (20) converts into classical Eq. (25) for transformation of the homocentric beam by the optical system. Examples of the calculation of the paraxial parameters of the two-element zoom system for transformation of Gaussian beam were given and the comparison to the classical two-element zoom system was also presented. Due to the fact that different lasers generate Gaussian beams with different parameters one would have to design special zoom lens for each type of laser. However, in practice it is more convenient to design a zoom lens for Gaussian beam of given parameters and the adaptation to the Gaussian beam with different parameters is performed by an appropriate optical system. Using Eq. (18) one can also solve other problems of the Gaussian beam transformation such as design of a laser beam expanders [29]. 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 [33]. Acknowledgments This work has been supported by the grant 13-31765S from the Czech Science Foundation. References [1] B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, 2nd Ed., WileyInterscience, New York, 2007. [2] A. Yariv, Optical Electronics, Oxford University Press, Oxford, 1990. [3] H. Kogelnik, T. Li, Laser beams and resonators, Appl. Opt. 5 (10) (1966) 1550–1567.
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[4] A.D. Clark, Zoom Lenses, Adam Hilger, London, 1973. [5] K. Yamaji, Progress in Optics, VI, North-Holland Publishing Co., Amsterdam, 1967. [6] A. Miks, J. Novak, P. Novak, Method of zoom lens design, Appl. Opt. 47 (32) (2008) 6088–6098. ´ [7] T. Kryszczynski, J. Mikucki, Structural optical design of the complex multigroup zoom systems by means of matrix optics, Opt. Express 21 (17) (2013) 19634–19647. [8] A. Wather, Angle eikonals for a perfect zoom system, J. Opt. Soc. Am. A 18 (8) (2001) 1968–1971. [9] G. Wooters, E.W. Silvertooth, Optically compensated zoom lens, J. Opt. Soc. Am. 55 (4) (1965) 347–351. [10] D.F. Kienholz, The design of a zoom Lens with a large computer, Appl. Opt. 9 (6) (1970) 1443–1452. [11] A.V. Grinkevich, Version of an objective with variable focal length, J. Opt. Technol. 73 (2006) 343–345. [12] K. Tanaka, Paraxial analysis of mechanically compensated zoom lenses. 1: fourcomponent type, Appl. Opt. 21 (12) (1982) 2174–2183. [13] G.H. Matter, E.T. Luszcz, A family of optically compensated zoom lenses, Appl. Opt. 9 (4) (1970) 844–848. [14] K. Tanaka, Erratum paraxial analysis of mechanically compensated zoom lenses. 1: four-component type, Appl. Opt. 21 (21) (1982) 3805. [15] K. Tanaka, Paraxial analysis of mechanically compensated zoom lenses. 2: generalization of Yamaji Type V, Appl. Opt. 21 (22) (1982) 4045–4053. [16] K. Tanaka, Paraxial analysis of mechanically compensated zoom lenses. 3: fivecomponent type, Appl. Opt. 22 (4) (1983) 541–553. ´ [17] T. Kryszczynski, M. Le´sniewski, J. Mikucki, Use of matrix optics to analyze the complex multi-group zoom systems, in: Proceedings of the SPIE 8697, SPIE, Bellingham, 2012, p. 86970I. [18] S. Pal, L. Hazra, Ab initio synthesis of linearly compensated zoom lenses by evolutionary programming, Appl. Opt. 50 (10) (2011) 1434–1441. [19] L.L. Hazra, S. Pal, A novel approach for structural synthesis of zoom systems, in: Proceedings of the SPIE 7786, SPIE, Bellingham, 2010, p. 778607. [20] S. Pal, L. Hazra, Stabilization of pupils in a zoom lens with two independent movements, Appl. Opt. 52 (23) (2013) 5611–5618. [21] S. Pal, Aberration correction of zoom lenses using evolutionary programming, Appl. Opt. 52 (23) (2013) 5724–5732. [22] A. Miks, Applied Optics, Czech Technical University Press, Prague, 2009. [23] S. Bollanti, P. Di Lazzaro, D. Murra, Performance of a zoom homogenizer for reshaping coherent laser beams, Optik Commun. 264 (1) (2006) 174–179. [24] F. Ge, S. Wang, D. Zhao, The representation of waist-to-waist transformation of Gaussian beams by the scaled fractional Fourier transform, Optik Laser Technol. 39 (4) (2007) 831–834. [25] P. Laures, Geometrical approach to Gaussian beam propagation, Appl. Opt. 6 (4) (1967) 747–755. [26] T.S. Chu, Geometrical representation of Gaussian beam propagation, Bell System Tech. J. 45 (1966) 287–299. [27] L.W. Casperson, Synthesis of Gaussian beam optical systems, Appl. Opt. 20 (13) (1981) 2243–2249. [28] L.C. Andrews, W.B. Miller, J.C. Ricklin, Geometrical representation of Gaussian beams propagating through complex paraxial optical systems, Appl. Opt. 32 (30) (1993) 5918–5929. [29] http://www.thorlabs.com/ . [30] W.T. Welford, Aberrations of the Symmetrical Optical Systems, Academic Press, London, 1974. [31] M. Herzberger, Modern Geometrical Optics, Interscience, New York, 1958. [32] W. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York, 2007. [33] A. Miks, J. Novak, Propagation of Gaussian beam in optical system with aberrations, Optik 114 (10) (2003) 437–440.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Paraxial properties of two-element zoom systems for Gaussian beam transformation Antonín Mikˇs, Pavel Novák ∗ Czech Technical University in Prague, Faculty of Civil Engineering, Department of Physics, Thakurova 7, 16629 Prague, Czech Republic
a r t i c l e
i n f o
Article history: Received 12 August 2014 Accepted 23 August 2015 Keywords: Zoom systems Geometric optics First-order optical design Gaussian beams
a b s t r a c t This work deals with a problem of paraxial analysis of optical systems for transformation of Gaussian beams. General equations describing the paraxial parameters of a two-element zoom system for transformation of a Gaussian beam are derived. It is shown that such a zoom lens has different kinematics with respect to classical zoom system designed for homocentric beams, which depends on the parameters of the transformed Gaussian beam. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Lasers [1–3] are currently used in many areas of science and technology. Different devices and instruments that take advantage of the properties of a laser beam are being developed nowadays. The laser beam used in these devices needs to be transformed using a special optical system to achieve the required properties and parameters of the output beam e.g., divergence, position of the beam waist etc. In many application areas (e.g., laser scanning, cutting, annealing, welding etc.) a continuous change of the laser beam parameters is required. This can be achieved by employing an optical system that makes possible a continuous change of some of its parameters (e.g., focal length and transverse magnification), which is usually referred to as a zoom lens. The problem of analysis and design of zoom lenses for classical (homocentric) light beams can be found e.g., in Refs. [4–21]. However, as it is well-known the laser beam is not homocentric, it is a Gaussian beam, therefore different equations hold for the transformation of such a beam through the optical system with respect to classical equations that are valid for homocentric light beams [1–3,22–28]. Thus one cannot use the well-known equations for the design of classical zoom lenses for homocentric light beams [4–22] to design a zoom system for transformation of a Gaussian beam. Different equations have to
∗ Corresponding author. Tel.: +420 224357919. E-mail address: [email protected] (P. Novák). http://dx.doi.org/10.1016/j.ijleo.2015.08.123 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
be derived that enable the paraxial design of the zoom lens for the transformation of Gaussian beam. In case the optical system made from classical optical elements makes possible to change the focal length in a continuous manner it has to have at least two members separated by a specific distance. By changing the mutual positions of those two elements one can achieve a continuous change of focal length of this system in a given range. The aim of our work is to derive equations for calculation of paraxial properties of the two-element zoom system for transforming a Gaussian beam. 2. Basic parameters of a Gaussian beam While the theory of Gaussian beams is well-known, we briefly outline it here for the benefit of the reader. Assume a harmonic electromagnetic wave field propagating through the homogenous isotropic medium. Such field is then described by the Helmholtz equation [1,2,22].
∇ 2 u(x, y, z) + k2 u(x, y, z) = 0, where, ∇ 2
(1)
is a Laplace operator, k = 2/ is the wave number, is the wavelength of light and u = u(x, y, z) is an arbitrary component of e.g., the electric field vector E [1,2,22] in orthogonal Cartesian coordinate system. Eq. (1) describes the electromagnetic field in the whole space. 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.
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A. Mikˇs, P. Novák / Optik 126 (2015) 4249–4253
In such a case the solution of Eq. (1) is sought in the following form [1,2,22]. (2) u(x, y, z) = ϕ(x, y, z) exp(−ikz), √ where, i = −1. By substitution of Eq. (2) into Eq. (1) we obtain 2
2
∂ ϕ ∂ ϕ ∂ϕ + − 2ik = 0, ∂x2 ∂ y2 ∂z
(3)
where we neglected the term ∂2 ϕ/∂z2 due to the reasons given above. Eq. (3) is a parabolic equation [1,2,22] for function ϕ(x, y, z). Let us now restrict our analysis on the most simple case of circular Gaussian beam, which is however the most important in practice. By substitution of solution of Eq. (3) into Eq. (2) one obtains [1,2,22–28] w0 exp w
u=
−
x 2 + y2 w2
exp
−ikz + i
− ik
x 2 + y2 2R
,
(4)
where, R = R(z) = z(1 + z02 /z 2 ) w2 = w2 (z) = w02 (1 + z 2 /z02 ) =
(z) = arctan(z/z0 )
,
(5)
z0 = kw02 /2 whereas, 2w0 denotes the diameter of the beam waist. As it is obvious from Eq. (4) the amplitude of the beam decrease in transverse direction with a Gaussian function, which is the reason for the denotation as Gaussian beam. In case that R1 and R2 are the radii of curvature of the mirrors in laser resonator and p is their mutual distance, it holds z02 =
p(R1 − p)(R2 − p)(R1 + R2 − p) (R1 + R2 − 2p)2
.
(6)
Equation for the wavefront of such Gaussian beam then according to Eq. (4) takes the form z+
x 2 + y2 − = z1 = konst. 2R k
(7)
Due to the fact that the wavelength emitted by e.g., He – Ne laser is very small (He–Ne = 0.0006328 mm) and the beam waist radius w0 of common He – Ne lasers is usually 0.3 – 0.5 mm [29], for z0 ≈ 1000 mm and z = 1000 mm the third term on the left hand side of the Eq. (7) is /k ≈ 0.00008, which can be neglected with respect to remaining two terms. Equation of the wavefront then can be written as x 2 + y2 = z1 − z. 2R
Fig. 1. Basic parameters of a Gaussian beam.
which is the equation of the one-sheeted rotational hyperboloid. The divergence of the beam is then characterized by the divergence angle , which is the angle between the asymptote of the x cross section of the hyperboloid and the axis of the beam (z-axis). The total angular spread of the beam is then given by angle 2. From Eq. (10) we obtain (tan ≈ ) w0 = /.
Fig. 1 shows schematically the basic parameters of the Gaussian beam and its transformation by optical system.
3. Transformation of a Gaussian beam by an optical system 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 will be valid for the thick lens system assuming we take all the values with respect to focal points F and F of individual optical elements (Fig. 2) or to their principal planes [22,30–32]. As it is known from the geometrical optics theory the thin lens (Fig. 2) transforms the incident spherical wavefront with radius R to output spherical wavefront with radius R according to the relation [22,30–32] 1/R − 1/R = 1/f ,
(8)
Eq. (8) is the equation of a rotational surface, whereas, R is its vertex radius of curvature i.e., it is the radius of curvature for the points in the close proximity to z-axis. From Eq. (5) we can find the minimum value of the radius of curvature Re by employing a condition for the extremum (dR/dz = 0) Re = 2z0 .
(9)
One can see that the radius of curvature changes from R =∞ for z = 0 to minimum value of R = Re for z = z0 and then it again increases to infinity for z→ ∞. The shape of the Gaussian beam in space is determined by the change of the amplitude. It is usual to describe the shape of the Gaussian beam by the surface (envelope) where the amplitude decrease to 1/e (e = 2.71828 is the Euler number) of the value on the beam axis. From Eq. (4) we have x2 + y2 = w02 (1 + z 2 /z02 ),
(10)
(11)
Fig. 2. Transformation of a Gaussian beam by the optical system.
(12)
A. Mikˇs, P. Novák / Optik 126 (2015) 4249–4253
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Fig. 3. Transformation of a Gaussian beam by two consecutive elements of a thin lens system. Fig. 4. Two element optical system (F and F are focal points and P and P are principal points of the system).
where, f is the focal length of the lens. Using the previous equation and Eq. (5) we obtain (Fig. 2) f
G=
2
q2 + (Re /2)
f
=
2
q
w0 =
2
q2 + z02
=
1 + m2 (z0 /f )
2
= −qG
Re = Re G
The transverse Gaussian beam waist magnification mG of the whole system is given by
m2
Re /k, w0 =
m2G =
.
Re /k
The meaning of individual symbols is evident from Fig. 2. The quantity mG is the Gaussian beam waist magnification (magnification of the beam waist) and m = f /q is the transverse magnification of the lens (optical system). Using Eq. (13) we obtain qq = −f
2
+ m2G (Re /2) = −f
2
For the case of the two-element optical system (Fig. 4), assuming that we know the focal lengths f1 , f2 and the distance d between the two lenses, we obtain the following relations using the equations from the previous section 2
q1 =
+ m2G z02
2
f − (Re /2) m2G
q = (1/mG ) q = −mG
2
2
.
(14)
Gi =
,
=
sF =
2
Eq. (14) is the generalization of the Newton’s lens formula for the Gaussian beam. For Re = z0 = 0 Eq. (14) becomes a standard Newton’s lens formula for homocentric beam. Let us now assume we have a thin lens system in air. Fig. 3 shows the situation for 2 consecutive lenses (ith and i + 1th lens). Using Eq. (13) then for the thin lens system of n lenses (i = 1, 2, 3,. . ., n) one obtains
Re,i+1 = Rei Gi
f 1 ±
2
q2
f − (Re /2) m2G
2 f i 2 2 qi + (Rei /2) qi = −qi Gi
(17)
4. Transformation of a Gaussian beam by the two-element zoom system
s = q − f , s = q + f
2
Gi .
i=1
(13)
w0 = w0 = / √ mG = w0 /w0 = G
i=n
(15)
i = fi + fi+1 − di
qi+1 = qi + i
m2G =
2
(f1 −
2
(f1 f2 /mG ) − (Re1 /2)
2
2 f 2 q1 (f 1 − q1 ) − Re1 /4 2
2
(f 1 − q1 ) + (Re1 /2)
−f1 (1
−
f1 /),
2 (f1 f 2 ) = 2 2 q1 ) + (Re1 /2) q = q1 − f1 − sF ,
sF
=
f2 (1 f
−
q2 + (Re1 /2)
q =
q2
+
f2
2
,
f2 /)
2 2
−
=
(18)
m2 1 + m2 (Re1 /2f )
2
sF
q = −qm2G , = f1 + f2 − d, f = f1 f2 /
where, f is the focal length of the two-element optical system and m = f /q is the transverse magnification of the system. The meaning of individual symbols if evident from Fig. 4. Now we will deal with the paraxial design of the zoom lens for Gaussian beams. Such system should hold fixed the distance L between the object A (beam waist of the incoming Gaussian beam) and the image A (beam waist of the output Gaussian beam) during the change of the magnification mG of the system (change of the diameter of the output beam waist). In case of such requirement, it holds (Fig. 4) −q − sF + d + sF + q = L = konst.
fi
2
(19)
Where, is the focal length of the ith lens and di is the distance between the ith and the i + 1th lens. In case we are interested in image and object axial distances we can calculate them as
By substitution of Eq. (18) into Eq. (19) we obtain for the parameter after tedious calculation the following equation
si = qi − fi , si = qi + fi .
4 + a3 3 + a2 2 + a1 + a0 = 0,
(16)
(20)
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A. Mikˇs, P. Novák / Optik 126 (2015) 4249–4253 Table 1 Two-element zoom system for Gaussian beam (solution 1).
where, 2
2
˛ = L − 2(f1 + f2 ), ˇ = f1 + f2 , = f1 f2 , ı = Re1 /2 a3 = 2˛
2
a2 = ˛2 + 2ˇ + ı(m2G + 1) a1 = 2˛ˇ
a0 = ˇ2 − (m2G + 1)/mG
f1 = 30, f2 = 25 w01 = 0.3, L = 20
.
(21)
2
For chosen focal lengths f1 and f2 of both members of the optical system, Gaussian beam waist magnification mG of the optical system, distance L between the object and the image (distance between the beam waists of the input and output beams), radius of Re1 /2z and the wavethe input Gaussian beam waist w01 = length of the input light beam entering the optical system, one can calculate the quantity by solving Eq. (20). The mutual distance d between the two lenses forming the zoom system can be calculated using d = f1 + f2 − .
(22)
By substitution of the calculated value into Eq. (18) one can obtain q1 and q2 . If we choose the magnification of the Gaussian
beam waist in an interval mG ∈ mG min , mG max then we obtain the mutual distance d = d(mG ) by the above described procedure using Eqs. (18), (20), (21) and (22). The axial distances s1 and s2 can then be determined by substitution of calculated q1 and q2 into relations (Fig. 4) s1 = q1 − f1 , s2 = q2 + f2 .
(23)
The transverse diameter of the Gaussian beam (i.e., the diameter given by the amplitude decrease to 1/e of the axial value) on individual members (lenses) of given two-element optical system can be calculated using Eq. (1). Thus the given problem of determination of paraxial parameters of the two-element zoom system is solved. In case of image axial distance s2 from the second lens of the twoelement system is negative, additional optical system with focal length f3 ≥ (−s1 + d)max has to be used. The position of the addi of the tional system is given by the requirement that the image w02 beam waist created by our two-element zoom system is placed in the object focal point of the additional optical system. The image of created by this additional optical system then the beam waist w02 would be located in the image focal point of this system. Re1 /2 of the In case we set the beam waist diameter w01 = incident Gaussian beam equal to zero i.e., we assume homocentric beam Eq. (20) then converts to the classical equation for a twoelement zoom system as it will be shown further. Eq. (20) can be modified to the form
( + ˛) + ˇ
2
− (m2G + 1)/mG
2
1 − (ımG )
2
= 0.
(24)
By setting Re1 = 0 i.e., ı = Re1 /2 = 0 in Eq. (24) (homocentric beam), we obtain 2 + ˛ + ˇ + (m2 + 1)/m = 0,
(25)
whereas, m = f /q (see Eq. (18)) is the transverse magnification of the optical system. Equation (25) then gives us the classical solution of the two-element zoom lens system. As it can be seen from Eqs. (20) and (25) zoom systems for the transformation of Gaussian beams have different kinematics of individual components with respect to zoom systems designed for classical homocentric beams.
mG 0.2 0.3 0.4 0.5 0.6 0.7 0.8
s1 −20.4782 −49.4240 −74.9422 −93.6256 −100.5899 −86.1914 −24.0868
D 46.8088 49.8700 51.5312 52.6068 53.3674 53.9343 54.3851
s2 −47.2869 −79.2940 −106.4733 −126.2324 −133.9574 −120.1257 −58.4719
Table 2 Two-element zoom system for Gaussian beam (solution 2). f1 = 30, f2 = 25, w01 = 0.3, L = 20 mG 0.2 0.3 0.4 0.5 0.6 0.7 0.8
s1 144.8605 180.7550 216.8562 250.9363 280.1627 298.6764 287.7910
d 61.9360 59.0084 57.4810 56.5407 55.9096 55.4608 55.1137
s2 102.9244 141.7466 179.3752 214.3957 244.2531 263.2155 252.6773
5. Examples of calculation of the two-element zoom system for Gaussian beam transformation In order to give the reader a rough idea of the differences in kinematics of individual components of the two-element zoom system for Gaussian and classical (homocentric) beams we will present the following example. Table 1 shows the results of the calculation of paraxial parameters of the two-element zoom lens for Gaussian beam transformation. As a laser we choose e.g., He – Ne laser HNL020R from Thorlabs [29] with beam waist radius w0 = 0.3 mm. Individual members of the optical system of the zoom lens are assumed to be thin lenses for simplicity. Parameters of the zoom system were chosen as f1 = 30 mm, f2 = 25 mm and L = 20 mm. By solving Eq. (20) and using Eqs. (18) and (23) we obtain for the first real root of the Eq. (20) the values given in Table 1. Table 2 then represents the results obtained for the second real root of Eq. (20). Table 3 then shows values calculated for homocentric light beam w01 = 0 for comparison. Values of length quantities in Tables 1–3 are given in millimeters. As it can be seen from Table 1 the axial distance of the image from the second element of the zoom lens s2 is negative therefore we will use additional optical system placed behind our zoom lens to image the beam waist. The focal length of the additional system can be chosen e.g., f3 = 160 mm and it is located in the distance f3 + L = 180 mm from the object plane of the zoom system (i.e., from the beam waist of the beam incident at the zoom lens). Image of the beam waist of the Gaussian beam will then be located of w03 at the image focal point F3 of this additional system i.e., at distance f3 = 160 mm from the additional system. As it can be seen further from Tables 2 and 3 the solution of the zoom for the Gaussian beam Table 3 Classical two-element zoom system for homocentric light beam. f1 = 30, f2 = 25, w01 = 0, L = 20 m −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8
s1 103.5990 105.7188 115.0806 130.6868 153.8083 187.8883 240.4160
d 76.3325 66.7891 61.7204 58.7340 56.9042 55.7868 55.1387
s2 47.2665 58.9297 73.3602 91.9528 116.9042 152.1015 205.2774
A. Mikˇs, P. Novák / Optik 126 (2015) 4249–4253
transformation differs significantly from the solution of the classical two-element zoom system for imaging by the homocentric light beam. 6. Conclusion In our work a detailed analysis of the paraxial transformation of the Gaussian beam by an optical system was performed. The equations for calculation of paraxial parameters of the two-element zoom system enabling continuous magnification of the beam waist diameter were derived. It was shown that the mutual distance of the two elements forming the two-element zoom system can be found by solving Eq. (20), which is a fourth order equation. It was shown that the zoom systems for Gaussian beam transformation have different kinematics of the motion of individual elements of zoom lens than the classical zoom systems for homocentric beams. Further it was shown that for the case of homocentric beam (Re1 = 0) the derived Eq. (20) converts into classical Eq. (25) for transformation of the homocentric beam by the optical system. Examples of the calculation of the paraxial parameters of the two-element zoom system for transformation of Gaussian beam were given and the comparison to the classical two-element zoom system was also presented. Due to the fact that different lasers generate Gaussian beams with different parameters one would have to design special zoom lens for each type of laser. However, in practice it is more convenient to design a zoom lens for Gaussian beam of given parameters and the adaptation to the Gaussian beam with different parameters is performed by an appropriate optical system. Using Eq. (18) one can also solve other problems of the Gaussian beam transformation such as design of a laser beam expanders [29]. 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 [33]. Acknowledgments This work has been supported by the grant 13-31765S from the Czech Science Foundation. References [1] B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, 2nd Ed., WileyInterscience, New York, 2007. [2] A. Yariv, Optical Electronics, Oxford University Press, Oxford, 1990. [3] H. Kogelnik, T. Li, Laser beams and resonators, Appl. Opt. 5 (10) (1966) 1550–1567.
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