- Email: [email protected]
Paraxial imaging properties of double conjugate zoom lens system composed of three tunable-focus lenses
Optics and Lasers in Engineering 53 (2014) 86–89
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Paraxial imaging properties of double conjugate zoom lens system composed of three tunable-focus lenses Antonín Mikš n, Jiří Novák Czech Technical University in Prague, Faculty of Civil Engineering, Department of Physics, Thakurova 7, 16629 Prague, Czech Republic
art ic l e i nf o
a b s t r a c t
Article history: Received 6 April 2013 Accepted 26 August 2013 Available online 19 September 2013
Novel optical systems based on tunable-focus lenses represent emerging lens technology, which will enable to design optical systems with variable optical parameters with no analogy in conventional systems. Our work describes paraxial imaging properties and design parameters of a double conjugate zoom lens composed of three tunable-focus lenses with a fixed position. General formulas for the calculation of optical power of individual tunable-focus lenses are derived. The application of derived formulas is presented on three examples of optical systems for different imaging conditions. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Zoom lens Tunable-focus lenses Optical imaging
1. Introduction Various active optical elements enabling a continuous change of their focal length have been developed and analyzed in recent years [1–22]. These modern and highly prospective optical elements [3,4] make possible to design absolutely novel optical systems, which have no analogy in classical optical system design. At present some commercially produced lenses with a tunable focal length are offered, which are based on various physical principles and the development of such lenses is rapidly increasing. However, their parameters are not still ideal and it is likely that various superior types of tunable-focus lenses will appear on the market in next years, from which various novel optical systems will be possible to design. For the purpose of design of such systems it is necessary to make a theoretical analysis of possibilities of the design of optical systems, which will use this new generation of active optical elements. The main advantage of such novel optical systems is simpler mechanical design due to the fact that individual elements have not to move mutually in order to achieve the change of optical parameters of the whole optical system. The change of focal length of individual elements can be realized by different physical principles, which are described in detail in Refs. [1–9], and we will not deal with them in this work. The technology of current fluidic tunable-focus lenses is usually based on electrowetting phenomena, the controlled injection of optical fluid into chambers with deformable membranes, thermooptical or electroactive polymers, and liquid crystals as active optical elements. The aim of this work is to analyze and calculate paraxial parameters of an optical system with variable magnification or focal length, which keeps the position of object and image planes and
n
Corresponding author. Tel.: þ 420 224354948. E-mail address: [email protected] (A. Mikš).
0143-8166/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlaseng.2013.08.018
entrance and exit pupil planes fixed and does not move its elements during the change of the magnification or focal length (double conjugate zoom lens system). Such an effect can be obtained using lenses with a tunable focal length. The double conjugate zoom lens system from tunable focus lenses satisfies the requirement that object, image and pupil planes are fixed during the change of magnification or focal length. Because of three conditions must be fulfilled the double conjugate zoom lens system from tunable focus lenses must be composed of at minimum three optical elements as the distances between individual elements must hold fixed with the magnification or focal length change. Formulas are derived, which enable to determine focal lengths of individual elements of the optical system in a general case, when an object is situated in a finite distance from the optical system. In work [22] formulas were derived for the calculation of values of optical power of individual elements of the three component optical system for the object in a finite distance, which are not too suitable for the further detailed analysis of the optical system. In this work novel equations are derived, which contain only external parameters of the optical system and are more suitable for the analysis of such optical systems. Equations for a special case, when the object is situated in infinity and for the case of a telescopic optical system, are derived in this work. The specific parameters of lenses (radii of curvature, index of refraction, etc.) are important during aberration design of the optical system. This problem is described in detail in Refs. [13–16] and we will not deal with aberration compensation in this work.
2. Paraxial properties of three-component double conjugate zoom lens Now, we will focus on an analysis of paraxial imaging properties of a three-component double conjugate zoom lens, which uses
A. Mikš, J. Novák / Optics and Lasers in Engineering 53 (2014) 86–89
Fig. 1. Scheme of three-component zoom lens composed of three tunable-focus lenses.
lenses with a continuously variable focal length. Consider that the optical system in Fig. 1 is composed of three thin lenses with a variable focal length (tunable focus lenses). It is required that the numerical values of distances s, s′, p, and p′ were constant with the change of the transverse magnification m ¼ y′=y. This means that positions of planes ξ, ξ′, η, and η′ are fixed during the change of magnification m (double conjugate zoom lens system: AA′ ¼ const:, PP′ ¼ const:). The meaning of other symbols is evident from Fig. 1. It is well-known that every optical system is characterized by its focal length f ′, position of the object focal point sF , and position of the image focal point s′F′ . The power φ of the optical system is defined by the formula φ ¼ n′=f ′, where n′ is the refractive index of image space. Further, if we consider that the image and object media is air (n ¼ n′ ¼ 1), the optical system is composed of N thin lenses, then we may define the following Gaussian brackets [23]
α ¼ ½dN1 ; φN1 ; dN2 ; φN2 ; dN3 ; φN3 ; :::; d1 ; φ1 ; β ¼ ½dN1 ; φN1 ; dN2 ; φN2 ; dN3 ; φN3 ; :::; d1 γ ¼ ½φN ; dN1 ; φN1 ; dN2 ; φN2 ; dN3 ; :::; d1 ; φ1 δ ¼ ½φN ; dN1 ; φN1 ; dN2 ; φN2 ; dN3 ; :::; d1
βαs 1 ; m ¼ s′γ þ α ¼ δγ s δsγ
1 mðspÞ½d2 ðp′s′Þ þ d1 mp′ þ d1 s′ðp′s′Þ þ d1 d1 ½m2 p′sðpsÞ þps′ðs′p′Þ
ð7Þ
φ2 ¼
d1 þ d2 ps′ p′sm þ þ d1 d2 mðpsÞ d1 d2 ðs′p′Þ d1 d2
ð8Þ
φ3 ¼
1 mðpsÞ½d1 ðs′p′Þ þ d2 ms þ d2 pðs′p′Þ þ d2 d2 ½m2 p′sðpsÞ þ ps′ðs′p′Þ
ð9Þ
Eqs. (7)–(9) represent a general solution of a given problem, when the object is situated at a finite distance s from the optical system. When one needs to calculate the optical power φ1 ; φ2 ; φ3 using formulas (7)–(9), the distances d1, d2 and external parameters s, s′, p, p′ and m must be chosen in advance. Now, we can calculate the form of Eqs. (7)–(9) for a special case, when the object is situated in infinity, i.e. s ¼ 1 and m¼ 0. Then, it holds lim ðmsÞ ¼ f ′
s-1
ð2Þ
where s is the distance of the object from the first element of the optical system, s′ is the distance of the image from the last element of the optical system, and m is the transverse magnification of the optical system. When we focus on the analysis of paraxial properties of a three-component zoom lens, values α; β ; γ , and δ are given by Eq. (1), φ1 ; φ2 ; φ3 are values of the optical power of individual components, and d1, d2 are distances between components of the zoom lens. Optical powers φ1 ; φ2 ; φ3 and distances d1, d2 are inner parameters of the optical system. Parameters s, s′, p, p′ and m are external parameters of the optical system. We can write for such zoom lens
α ¼ 1d2 ðφ1 þ φ2 φ1 φ2 d1 Þφ1 d1 ;
ð3Þ
β ¼ d1 þ d2 φ2 d1 d2
ð4Þ
γ ¼ φ ¼ ðφ1 þ φ2 þ φ3 Þ þ φ1 φ2 d1 þ φ2 φ3 d2 þ φ1 φ3 ðd1 þ d2 Þφ1 φ2 φ3 d1 d2 ;
ð5Þ
δ ¼ 1d1 ðφ2 þ φ3 Þd2 φ3 þ d1 d2 φ2 φ3
ð6Þ
Using formulas (3)–(6) one can derive for values of the optical power of individual components of the optical system after a tedious calculation the following formulas
ð10Þ
we obtain using previous equations for values of the optical power of individual elements
φ1 ¼
1 φðs′p′Þðd2 þ φd1 s′Þ þ d1 d1 ½pφ2 s′ðp′s′Þ þ p′
ð11Þ
φ2 ¼
d1 þ d2 φps′p′=φðp′s′Þ d1 d2
ð12Þ
φ3 ¼
1 d2 þ φðs′p′Þðd1 φd2 pÞ þ d2 d2 ½pφ2 s′ðp′s′Þ þp′
ð13Þ
Formulas (11)–(13) enable to calculate the optical power of individual elements of the optical system in the case, when the object is in infinity. Assume the case of a telescopic optical system, where it holds that φ ¼ 0 and s ¼ s′ ¼ 1. We may consider that it holds lim ðφs′Þ ¼ m
where φi is the power of i-th lens, and di is the distance between (iþ 1)-st a i-th lens. Then, it holds for fundamental paraxial parameters [23]
φ ¼ γ ; sF ¼ δ=γ ; s′F′ ¼ α=γ ; s′ ¼
φ1 ¼
s′-1
ð1Þ
87
ð14Þ
Then, we obtain from Eqs. (11)–(13) for optical powers of individual elements
φ1 ¼
1 mðmd1 þd2 Þ þ d1 d1 ðp′m2 pÞ
ð15Þ
φ2 ¼
d1 þ d2 þ p′=mmp d1 d2
ð16Þ
φ3 ¼
1 md1 þ d2 þ d2 d2 ðp′m2 pÞ
ð17Þ
Formulas (7)–(17) represent a complex solution of the problem of the paraxial analysis of the three-component double conjugate zoom lens, which uses lenses with a continuously variable magnification or focal length. 3. Examples of three-component double conjugate zoom lens We will present the application of above-mentioned equations on several examples of optical systems. In the first case we have the optical system with a variable magnification and the object is situated in a finite distance in front of the optical system (e.g. microscope objective lens, etc.), the second example presents the optical system with a variable focal length and the object situated at infinity (e.g. photographic objective lens, etc.), and the third case represents the telescopic afocal system, when the object and image are situated at infinity (e.g. riflescope, etc.). Table 1 describes an example of optical system of the objective lens with a variable magnification ðm A 〈2; 10〉Þ, a constant position of the image plane and the exit pupil plane. Values hi
88
A. Mikš, J. Novák / Optics and Lasers in Engineering 53 (2014) 86–89
Table 1 Calculated parameters—zoom lens ðm A 〈2; 10〉Þ d1 ¼ 30, d2 ¼ 30, s ¼ 20, s′ ¼ 115, p ¼ 1, p′ ¼ 30 m
f ′1
f ′2
f ′3
h1
h2
h3
y1
y2
y3
2 4 6 8 10
40.9498 33.8676 25.7602 21.2168 18.6747
155.8519 61.2293 56.2285 71.8744 126.3882
46.4308 217.4707 226.2354 119.9412 102.3447
5.0 5.0 5.0 5.0 5.0
8.8370 8.0710 6.6771 5.4301 4.4677
14.3750 7.1875 4.7917 3.5938 2.8750
5.0000 2.5000 1.6667 1.2500 1.0000
1.3368 0.2854 0.2744 0.5175 0.6065
2.0690 2.0690 2.0690 2.0690 2.0690
Table 2 Calculated parameters—zoom lens with variable focal length ðf ′ A 〈100; 500〉Þ. d1 ¼ 100, d2 ¼ 100, s′ ¼ 90, p ¼ 1, p′ ¼ 50, F ¼ 5:6 f
f ′1
f ′2
f ′3
h1
h2
h3
y1
y2
y3
100 200 300 400 500
39.5097 63.2493 74.2421 80.3142 84.1146
30.8547 22.2445 17.4004 14.2903 12.1239
26.2312 29.3917 30.6036 31.2452 31.6425
8.9286 17.8571 26.7857 35.7143 44.6429
13.6699 10.3758 9.2931 8.7539 8.4310
8.0357 8.0357 8.0357 8.0357 8.0357
0.0500 0.0250 0.0167 0.0125 0.0100
5.0766 2.5145 1.6724 1.2531 1.0019
6.2500 6.2500 6.2500 6.2500 6.2500
Table 3 Telescopic optical system with a variable magnification (Γ A 〈4; 12〉). d1 ¼ 100, d2 ¼ 100, p ¼ 1, p′ ¼ 40 Γ
f ′1
f ′2
f ′3
h1
h2
h3
y1
y2
y3
4 6 8 10 12
56.1403 67.2897 73.9884 78.4314 81.5864
27.7778 22.7273 19.2308 16.6667 14.7059
24.2424 25.5319 26.2295 26.6667 26.9663
15.0 15.0 15.0 15.0 15.0
11.7188 7.2917 5.2734 4.1250 3.3854
3.7500 2.5000 1.8750 1.5000 1.2500
0 0 0 0 0
5.1563 3.2639 2.3828 1.8750 1.5451
8.2500 7.8333 7.6250 7.5000 7.4167
and yi (i¼1,2,3) in Table 1 denote incidence heights of the aperture and chief ray at the individual optical elements. The numerical aperture of the objective lens is NA ¼0.25. Table 2 presents an example of the calculation of parameters of the objective lens with a variable focal length ðf ′ A 〈100; 500〉Þ, constant position of the image plane and exit pupil plane. Values hi and yi (i¼1,2,3) in Table 2 denote incidence heights of the aperture and chief ray at the individual optical elements. The f-number of the objective lens is F¼ 5.6. Table 3 presents an example of the calculation of the telescopic optical system with a variable magnification and the constant position of the exit pupil for the angular magnification Γ ¼ tan ω′= tan ω from the range Γ A 〈4; 12〉. Values hi and yi (i¼1,2,3) in Table 3 denote incidence heights of the aperture and chief ray at the individual optical elements for the diameter of the field of view 10 mm in the object plane of the third element (eyepiece). The diameter of the entrance pupil of the optical system is 30 mm.
4. Conclusion This work presented a theoretical analysis of paraxial imaging properties and design parameters of a double conjugate zoom lens with variable magnification or focal length composed of three tunable-focus lenses. Such a system keeps the position of object and image planes and entrance and exit pupil planes fixed and it does not move its elements during the change of the magnification or focal length. The distances between individual elements are fixed during the magnification or focal length change. General formulas for the calculation of optical power of individual tunable-focus lenses are
derived, which are applicable to different imaging conditions. The calculation of paraxial parameters using the derived formulas were performed on three examples for different imaging conditions, namely the lens with a variable magnification for the object situated in a finite distance, the lens with a variable focal length for the object situated in an infinite distance, and a telescopic optical system. The derived formulas can be used for the initial design of such optical systems composed of three tunable-focus lenses.
Acknowledgment This work has been supported by the grant 13–31765S from the Czech Science Foundation.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Li G. Adaptive lens. Prog. Opt. 2010;55:199–284. Ren H, Wu ST. Introduction to Adaptive Lenses. Wiley; 2012. 〈http://www.varioptic.com/〉. 〈http://www.optotune.com/〉. Berge B, Peseux J. Variable focal lens controlled by an external voltage: an application of electrowetting. Eur. Phys. J. E 2000;3:159–63. Kuiper S, Hendriks BHW. Variable-focus liquid lens for miniature cameras. Appl. Phys. Lett. 2004;85:1128–30. Peng R, Chen J, Zhuang S. Electrowetting–actuated zoom lens with sphericalinterface liquid lenses. J. Opt. Soc. Am. A 2008;25:2644–50. Ren HW, Wu ST. Variable-focus liquid lens. Opt. Express 2007;15:5931–6. Beadie G, Sandrock ML, Wiggins MJ, Lepkowicz RS, Shirk JS, Ponting M, et al. Tunable polymer lens. Opt. Express 2008;16:11847–57. Reichelt S, Zappe H. Design of spherically corrected, achromatic variable- focus liquid lenses. Opt. Express 2007;15:14146–54.
A. Mikš, J. Novák / Optics and Lasers in Engineering 53 (2014) 86–89
[11] Li L, Wang QH, Jiang W. Liquid lens with double tunable surfaces for large power tunability and improved optical performance. J. Opt 2011;13:115503. [12] Li L, Wang QH. Zoom lens design using liquid lenses for achromatic and spherical aberration corrected target. Opt. Eng 2012;51:043001. [13] Mikš A, Novák J, Novák P. Generalized refractive tunable-focus lens and its imaging characteristics. Opt. Express 2010;18:9034–47. [14] Mikš A, Novák J. Analysis of two-element zoom systems based on variable power lenses. Opt. Express 2010;18:6797–810. [15] Mikš A, Novák J. Analysis of three-element zoom lens based on refractive variable-focus lenses. Opt. Express 2011;19:23989–96. [16] Mikš A, Novák J. Third-order aberrations of the thin refractive tunable-focus lens. Opt. Lett 2010;35:1031–3. [17] Mikš A, Novák J, Novák P. Three-element zoom lens with fixed distance between focal points. Opt. Lett 2012;37:2187–9.
89
[18] Valley P, Mathine DL, Dodge MR, Schwiegerling J, Peyman G, Peyghambarian N. Tunable-focus flat liquid-crystal diffractive lens. Opt. Lett. 2010;35 (3):336–8. [19] Chang J-H, Jung K-D, Lee E, Choi M, Lee S, Kim W. Varifocal liquid lens based on microelectrofluidic technology. Opt. Lett. 2012;37:4377–9. [20] Fuh Y-K, Lin M-X, Lee S. Characterizing aberration of a pressure-actuated tunable biconvex microlens with a simple spherically-corrected design. Opt. Laser Eng 2012;50:1677–82. [21] Tsai FS, Hwan Cho S, Lo Y-H, Vasko B, Vasko J. Miniaturized universal imaging device using fluidic lens. Opt. Lett. 2008;33:291–3. [22] Mikš A, Novák J. Three-component double conjugate zoom lens system from tunable focus lenses. Appl. Opt. 2013;52:862–5. [23] Herzberger M. Modern Geometrical Optics. New York: Interscience; 1958.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Paraxial imaging properties of double conjugate zoom lens system composed of three tunable-focus lenses Antonín Mikš n, Jiří Novák Czech Technical University in Prague, Faculty of Civil Engineering, Department of Physics, Thakurova 7, 16629 Prague, Czech Republic
art ic l e i nf o
a b s t r a c t
Article history: Received 6 April 2013 Accepted 26 August 2013 Available online 19 September 2013
Novel optical systems based on tunable-focus lenses represent emerging lens technology, which will enable to design optical systems with variable optical parameters with no analogy in conventional systems. Our work describes paraxial imaging properties and design parameters of a double conjugate zoom lens composed of three tunable-focus lenses with a fixed position. General formulas for the calculation of optical power of individual tunable-focus lenses are derived. The application of derived formulas is presented on three examples of optical systems for different imaging conditions. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Zoom lens Tunable-focus lenses Optical imaging
1. Introduction Various active optical elements enabling a continuous change of their focal length have been developed and analyzed in recent years [1–22]. These modern and highly prospective optical elements [3,4] make possible to design absolutely novel optical systems, which have no analogy in classical optical system design. At present some commercially produced lenses with a tunable focal length are offered, which are based on various physical principles and the development of such lenses is rapidly increasing. However, their parameters are not still ideal and it is likely that various superior types of tunable-focus lenses will appear on the market in next years, from which various novel optical systems will be possible to design. For the purpose of design of such systems it is necessary to make a theoretical analysis of possibilities of the design of optical systems, which will use this new generation of active optical elements. The main advantage of such novel optical systems is simpler mechanical design due to the fact that individual elements have not to move mutually in order to achieve the change of optical parameters of the whole optical system. The change of focal length of individual elements can be realized by different physical principles, which are described in detail in Refs. [1–9], and we will not deal with them in this work. The technology of current fluidic tunable-focus lenses is usually based on electrowetting phenomena, the controlled injection of optical fluid into chambers with deformable membranes, thermooptical or electroactive polymers, and liquid crystals as active optical elements. The aim of this work is to analyze and calculate paraxial parameters of an optical system with variable magnification or focal length, which keeps the position of object and image planes and
n
Corresponding author. Tel.: þ 420 224354948. E-mail address: [email protected] (A. Mikš).
0143-8166/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlaseng.2013.08.018
entrance and exit pupil planes fixed and does not move its elements during the change of the magnification or focal length (double conjugate zoom lens system). Such an effect can be obtained using lenses with a tunable focal length. The double conjugate zoom lens system from tunable focus lenses satisfies the requirement that object, image and pupil planes are fixed during the change of magnification or focal length. Because of three conditions must be fulfilled the double conjugate zoom lens system from tunable focus lenses must be composed of at minimum three optical elements as the distances between individual elements must hold fixed with the magnification or focal length change. Formulas are derived, which enable to determine focal lengths of individual elements of the optical system in a general case, when an object is situated in a finite distance from the optical system. In work [22] formulas were derived for the calculation of values of optical power of individual elements of the three component optical system for the object in a finite distance, which are not too suitable for the further detailed analysis of the optical system. In this work novel equations are derived, which contain only external parameters of the optical system and are more suitable for the analysis of such optical systems. Equations for a special case, when the object is situated in infinity and for the case of a telescopic optical system, are derived in this work. The specific parameters of lenses (radii of curvature, index of refraction, etc.) are important during aberration design of the optical system. This problem is described in detail in Refs. [13–16] and we will not deal with aberration compensation in this work.
2. Paraxial properties of three-component double conjugate zoom lens Now, we will focus on an analysis of paraxial imaging properties of a three-component double conjugate zoom lens, which uses
A. Mikš, J. Novák / Optics and Lasers in Engineering 53 (2014) 86–89
Fig. 1. Scheme of three-component zoom lens composed of three tunable-focus lenses.
lenses with a continuously variable focal length. Consider that the optical system in Fig. 1 is composed of three thin lenses with a variable focal length (tunable focus lenses). It is required that the numerical values of distances s, s′, p, and p′ were constant with the change of the transverse magnification m ¼ y′=y. This means that positions of planes ξ, ξ′, η, and η′ are fixed during the change of magnification m (double conjugate zoom lens system: AA′ ¼ const:, PP′ ¼ const:). The meaning of other symbols is evident from Fig. 1. It is well-known that every optical system is characterized by its focal length f ′, position of the object focal point sF , and position of the image focal point s′F′ . The power φ of the optical system is defined by the formula φ ¼ n′=f ′, where n′ is the refractive index of image space. Further, if we consider that the image and object media is air (n ¼ n′ ¼ 1), the optical system is composed of N thin lenses, then we may define the following Gaussian brackets [23]
α ¼ ½dN1 ; φN1 ; dN2 ; φN2 ; dN3 ; φN3 ; :::; d1 ; φ1 ; β ¼ ½dN1 ; φN1 ; dN2 ; φN2 ; dN3 ; φN3 ; :::; d1 γ ¼ ½φN ; dN1 ; φN1 ; dN2 ; φN2 ; dN3 ; :::; d1 ; φ1 δ ¼ ½φN ; dN1 ; φN1 ; dN2 ; φN2 ; dN3 ; :::; d1
βαs 1 ; m ¼ s′γ þ α ¼ δγ s δsγ
1 mðspÞ½d2 ðp′s′Þ þ d1 mp′ þ d1 s′ðp′s′Þ þ d1 d1 ½m2 p′sðpsÞ þps′ðs′p′Þ
ð7Þ
φ2 ¼
d1 þ d2 ps′ p′sm þ þ d1 d2 mðpsÞ d1 d2 ðs′p′Þ d1 d2
ð8Þ
φ3 ¼
1 mðpsÞ½d1 ðs′p′Þ þ d2 ms þ d2 pðs′p′Þ þ d2 d2 ½m2 p′sðpsÞ þ ps′ðs′p′Þ
ð9Þ
Eqs. (7)–(9) represent a general solution of a given problem, when the object is situated at a finite distance s from the optical system. When one needs to calculate the optical power φ1 ; φ2 ; φ3 using formulas (7)–(9), the distances d1, d2 and external parameters s, s′, p, p′ and m must be chosen in advance. Now, we can calculate the form of Eqs. (7)–(9) for a special case, when the object is situated in infinity, i.e. s ¼ 1 and m¼ 0. Then, it holds lim ðmsÞ ¼ f ′
s-1
ð2Þ
where s is the distance of the object from the first element of the optical system, s′ is the distance of the image from the last element of the optical system, and m is the transverse magnification of the optical system. When we focus on the analysis of paraxial properties of a three-component zoom lens, values α; β ; γ , and δ are given by Eq. (1), φ1 ; φ2 ; φ3 are values of the optical power of individual components, and d1, d2 are distances between components of the zoom lens. Optical powers φ1 ; φ2 ; φ3 and distances d1, d2 are inner parameters of the optical system. Parameters s, s′, p, p′ and m are external parameters of the optical system. We can write for such zoom lens
α ¼ 1d2 ðφ1 þ φ2 φ1 φ2 d1 Þφ1 d1 ;
ð3Þ
β ¼ d1 þ d2 φ2 d1 d2
ð4Þ
γ ¼ φ ¼ ðφ1 þ φ2 þ φ3 Þ þ φ1 φ2 d1 þ φ2 φ3 d2 þ φ1 φ3 ðd1 þ d2 Þφ1 φ2 φ3 d1 d2 ;
ð5Þ
δ ¼ 1d1 ðφ2 þ φ3 Þd2 φ3 þ d1 d2 φ2 φ3
ð6Þ
Using formulas (3)–(6) one can derive for values of the optical power of individual components of the optical system after a tedious calculation the following formulas
ð10Þ
we obtain using previous equations for values of the optical power of individual elements
φ1 ¼
1 φðs′p′Þðd2 þ φd1 s′Þ þ d1 d1 ½pφ2 s′ðp′s′Þ þ p′
ð11Þ
φ2 ¼
d1 þ d2 φps′p′=φðp′s′Þ d1 d2
ð12Þ
φ3 ¼
1 d2 þ φðs′p′Þðd1 φd2 pÞ þ d2 d2 ½pφ2 s′ðp′s′Þ þp′
ð13Þ
Formulas (11)–(13) enable to calculate the optical power of individual elements of the optical system in the case, when the object is in infinity. Assume the case of a telescopic optical system, where it holds that φ ¼ 0 and s ¼ s′ ¼ 1. We may consider that it holds lim ðφs′Þ ¼ m
where φi is the power of i-th lens, and di is the distance between (iþ 1)-st a i-th lens. Then, it holds for fundamental paraxial parameters [23]
φ ¼ γ ; sF ¼ δ=γ ; s′F′ ¼ α=γ ; s′ ¼
φ1 ¼
s′-1
ð1Þ
87
ð14Þ
Then, we obtain from Eqs. (11)–(13) for optical powers of individual elements
φ1 ¼
1 mðmd1 þd2 Þ þ d1 d1 ðp′m2 pÞ
ð15Þ
φ2 ¼
d1 þ d2 þ p′=mmp d1 d2
ð16Þ
φ3 ¼
1 md1 þ d2 þ d2 d2 ðp′m2 pÞ
ð17Þ
Formulas (7)–(17) represent a complex solution of the problem of the paraxial analysis of the three-component double conjugate zoom lens, which uses lenses with a continuously variable magnification or focal length. 3. Examples of three-component double conjugate zoom lens We will present the application of above-mentioned equations on several examples of optical systems. In the first case we have the optical system with a variable magnification and the object is situated in a finite distance in front of the optical system (e.g. microscope objective lens, etc.), the second example presents the optical system with a variable focal length and the object situated at infinity (e.g. photographic objective lens, etc.), and the third case represents the telescopic afocal system, when the object and image are situated at infinity (e.g. riflescope, etc.). Table 1 describes an example of optical system of the objective lens with a variable magnification ðm A 〈2; 10〉Þ, a constant position of the image plane and the exit pupil plane. Values hi
88
A. Mikš, J. Novák / Optics and Lasers in Engineering 53 (2014) 86–89
Table 1 Calculated parameters—zoom lens ðm A 〈2; 10〉Þ d1 ¼ 30, d2 ¼ 30, s ¼ 20, s′ ¼ 115, p ¼ 1, p′ ¼ 30 m
f ′1
f ′2
f ′3
h1
h2
h3
y1
y2
y3
2 4 6 8 10
40.9498 33.8676 25.7602 21.2168 18.6747
155.8519 61.2293 56.2285 71.8744 126.3882
46.4308 217.4707 226.2354 119.9412 102.3447
5.0 5.0 5.0 5.0 5.0
8.8370 8.0710 6.6771 5.4301 4.4677
14.3750 7.1875 4.7917 3.5938 2.8750
5.0000 2.5000 1.6667 1.2500 1.0000
1.3368 0.2854 0.2744 0.5175 0.6065
2.0690 2.0690 2.0690 2.0690 2.0690
Table 2 Calculated parameters—zoom lens with variable focal length ðf ′ A 〈100; 500〉Þ. d1 ¼ 100, d2 ¼ 100, s′ ¼ 90, p ¼ 1, p′ ¼ 50, F ¼ 5:6 f
f ′1
f ′2
f ′3
h1
h2
h3
y1
y2
y3
100 200 300 400 500
39.5097 63.2493 74.2421 80.3142 84.1146
30.8547 22.2445 17.4004 14.2903 12.1239
26.2312 29.3917 30.6036 31.2452 31.6425
8.9286 17.8571 26.7857 35.7143 44.6429
13.6699 10.3758 9.2931 8.7539 8.4310
8.0357 8.0357 8.0357 8.0357 8.0357
0.0500 0.0250 0.0167 0.0125 0.0100
5.0766 2.5145 1.6724 1.2531 1.0019
6.2500 6.2500 6.2500 6.2500 6.2500
Table 3 Telescopic optical system with a variable magnification (Γ A 〈4; 12〉). d1 ¼ 100, d2 ¼ 100, p ¼ 1, p′ ¼ 40 Γ
f ′1
f ′2
f ′3
h1
h2
h3
y1
y2
y3
4 6 8 10 12
56.1403 67.2897 73.9884 78.4314 81.5864
27.7778 22.7273 19.2308 16.6667 14.7059
24.2424 25.5319 26.2295 26.6667 26.9663
15.0 15.0 15.0 15.0 15.0
11.7188 7.2917 5.2734 4.1250 3.3854
3.7500 2.5000 1.8750 1.5000 1.2500
0 0 0 0 0
5.1563 3.2639 2.3828 1.8750 1.5451
8.2500 7.8333 7.6250 7.5000 7.4167
and yi (i¼1,2,3) in Table 1 denote incidence heights of the aperture and chief ray at the individual optical elements. The numerical aperture of the objective lens is NA ¼0.25. Table 2 presents an example of the calculation of parameters of the objective lens with a variable focal length ðf ′ A 〈100; 500〉Þ, constant position of the image plane and exit pupil plane. Values hi and yi (i¼1,2,3) in Table 2 denote incidence heights of the aperture and chief ray at the individual optical elements. The f-number of the objective lens is F¼ 5.6. Table 3 presents an example of the calculation of the telescopic optical system with a variable magnification and the constant position of the exit pupil for the angular magnification Γ ¼ tan ω′= tan ω from the range Γ A 〈4; 12〉. Values hi and yi (i¼1,2,3) in Table 3 denote incidence heights of the aperture and chief ray at the individual optical elements for the diameter of the field of view 10 mm in the object plane of the third element (eyepiece). The diameter of the entrance pupil of the optical system is 30 mm.
4. Conclusion This work presented a theoretical analysis of paraxial imaging properties and design parameters of a double conjugate zoom lens with variable magnification or focal length composed of three tunable-focus lenses. Such a system keeps the position of object and image planes and entrance and exit pupil planes fixed and it does not move its elements during the change of the magnification or focal length. The distances between individual elements are fixed during the magnification or focal length change. General formulas for the calculation of optical power of individual tunable-focus lenses are
derived, which are applicable to different imaging conditions. The calculation of paraxial parameters using the derived formulas were performed on three examples for different imaging conditions, namely the lens with a variable magnification for the object situated in a finite distance, the lens with a variable focal length for the object situated in an infinite distance, and a telescopic optical system. The derived formulas can be used for the initial design of such optical systems composed of three tunable-focus lenses.
Acknowledgment This work has been supported by the grant 13–31765S from the Czech Science Foundation.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Li G. Adaptive lens. Prog. Opt. 2010;55:199–284. Ren H, Wu ST. Introduction to Adaptive Lenses. Wiley; 2012. 〈http://www.varioptic.com/〉. 〈http://www.optotune.com/〉. Berge B, Peseux J. Variable focal lens controlled by an external voltage: an application of electrowetting. Eur. Phys. J. E 2000;3:159–63. Kuiper S, Hendriks BHW. Variable-focus liquid lens for miniature cameras. Appl. Phys. Lett. 2004;85:1128–30. Peng R, Chen J, Zhuang S. Electrowetting–actuated zoom lens with sphericalinterface liquid lenses. J. Opt. Soc. Am. A 2008;25:2644–50. Ren HW, Wu ST. Variable-focus liquid lens. Opt. Express 2007;15:5931–6. Beadie G, Sandrock ML, Wiggins MJ, Lepkowicz RS, Shirk JS, Ponting M, et al. Tunable polymer lens. Opt. Express 2008;16:11847–57. Reichelt S, Zappe H. Design of spherically corrected, achromatic variable- focus liquid lenses. Opt. Express 2007;15:14146–54.
A. Mikš, J. Novák / Optics and Lasers in Engineering 53 (2014) 86–89
[11] Li L, Wang QH, Jiang W. Liquid lens with double tunable surfaces for large power tunability and improved optical performance. J. Opt 2011;13:115503. [12] Li L, Wang QH. Zoom lens design using liquid lenses for achromatic and spherical aberration corrected target. Opt. Eng 2012;51:043001. [13] Mikš A, Novák J, Novák P. Generalized refractive tunable-focus lens and its imaging characteristics. Opt. Express 2010;18:9034–47. [14] Mikš A, Novák J. Analysis of two-element zoom systems based on variable power lenses. Opt. Express 2010;18:6797–810. [15] Mikš A, Novák J. Analysis of three-element zoom lens based on refractive variable-focus lenses. Opt. Express 2011;19:23989–96. [16] Mikš A, Novák J. Third-order aberrations of the thin refractive tunable-focus lens. Opt. Lett 2010;35:1031–3. [17] Mikš A, Novák J, Novák P. Three-element zoom lens with fixed distance between focal points. Opt. Lett 2012;37:2187–9.
89
[18] Valley P, Mathine DL, Dodge MR, Schwiegerling J, Peyman G, Peyghambarian N. Tunable-focus flat liquid-crystal diffractive lens. Opt. Lett. 2010;35 (3):336–8. [19] Chang J-H, Jung K-D, Lee E, Choi M, Lee S, Kim W. Varifocal liquid lens based on microelectrofluidic technology. Opt. Lett. 2012;37:4377–9. [20] Fuh Y-K, Lin M-X, Lee S. Characterizing aberration of a pressure-actuated tunable biconvex microlens with a simple spherically-corrected design. Opt. Laser Eng 2012;50:1677–82. [21] Tsai FS, Hwan Cho S, Lo Y-H, Vasko B, Vasko J. Miniaturized universal imaging device using fluidic lens. Opt. Lett. 2008;33:291–3. [22] Mikš A, Novák J. Three-component double conjugate zoom lens system from tunable focus lenses. Appl. Opt. 2013;52:862–5. [23] Herzberger M. Modern Geometrical Optics. New York: Interscience; 1958.