Paraxial analysis of three-component zoom lens with fixed distance between object and image points and fixed position of image-space focal point

Abstract

This work performs an analysis of basic optical properties of zoom lenses with a fixed distance between object and image points and a fixed position of the image-space focal point. Formulas for the calculation of paraxial parameters of such optical systems are derived and the calculation is presented on examples.

Full text

Paraxial analysis of three-component zoom lens
with fixed distance between object and image
points and fixed position of image-space
focal point
Antonin Miks* and Jiri Novak
Czech Technical University in Prague, Faculty of Civil Engineering, Department of Physics, Thakurova 7, 16629
Prague, Czech Republic
*miks@fsv.cvut.cz
Abstract: This work performs an analysis of basic optical properties of
zoom lenses with a fixed distance between object and image points and a
fixed position of the image-space focal point. Formulas for the calculation
of paraxial parameters of such optical systems are derived and the
calculation is presented on examples.
©2014 Optical Society of America
OCIS codes: (080.2468) First-order optics; (080.2740) Geometric optical design; (080.3630)
Lenses; (080.3620) Lens system design; (220.3630) Lenses.
References and links
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170.
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of matrix optics,” Opt. Express 21(17), 19634–19647 (2013).
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21(12), 2174–2183 (1982).
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(1970).
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matrix optics,” Proc. SPIE 7141, 71411Y (2008).
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zoom systems,” Proc. SPIE 8697, 86970I (2012).
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(2010).
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(2013).
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Received 16 Apr 2014; accepted 11 Jun 2014; published 18 Jun 2014
(C) 2014 OSA
30 June 2014 | Vol. 22, No. 13 | DOI:10.1364/OE.22.015571 | OPTICS EXPRESS 15571

24. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
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1871 (2002).
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systems,” Appl. Opt. 51(30), 7286–7294 (2012).
1. Introduction
Optical systems with variable optical parameters (zoom lenses) find applications in various
areas and many papers [1–22] are dedicated to their optical design. The change of a focal
length or a transverse magnification can be achieved by the change of the position of
individual elements of the optical system. The position of the image plane is required to be
fixed during the change of the focal length in order the image was sharp in the whole range of
the focal length change. If it is required that the image should be located at a specific constant
distance from the object for a given range of magnification, then the position of the image-
space focal point of the classical zoom lens is not fixed and changes its position during the
magnification change.
The aim of this work is to perform an analysis of paraxial optical properties of zoom
lenses with a fixed distance between object and image points and a fixed position of the
image-space focal point and derive formulas for the calculation of paraxial parameters of such
optical systems. Zoom lenses with the fixed distance between object and image points and
fixed position of the image-space focal point may find their applications in optical systems for
information processing [23], where it is possible to affect the amplitude and phase using the
spatial filter, which is positioned in the focal plane. As far as we know the analysis of such a
type of the zoom lens was not published yet.
2. Paraxial imaging properties of optical system
It is well-known from the theory of geometrical optics that every optical system is
characterized by its focal length
f
′, the position of the object focal point
F
s
, and the position
of the image focal point
F
s
′
′. The power φ of the optical system is defined by the formula
φ/nf
′′
=, where n′ is the refractive index of image space. Further, if we consider that the
image and object media is air ( 1nn
′
==) and the optical system is composed of N thin lenses,
then we can define Gaussian brackets [24, 25]
[
]
[]
[]
[]
112 23311
112 2331
112 2311
112 231
,φ,,φ,,φ,..., , φ,
,φ,,φ,,φ,..., ,
φ,,φ,,φ, ,..., , φ,
φ,,φ,,φ, ,..., ,
NNN NNN
NNN NNN
NN N N N N
NN N N N N
dd d d
dd d d
dd dd
dd dd
α
β
γ
δ
−−− −−−
−−− −− −
−−− −−
−−− −−
=− − − −
=− − −
=− − − −
=− − −
(1)
where φi is the power of i-th lens, and i
dis the distance between (i + 1)-st a i-th lens. Then, it
holds for basic paraxial parameters [24, 25]
1
φ,/, /, , ,
FF
s
ss s ms
s
s
β
α
γδγ αγ γα
δ
γ
δ
γ
′−
′′ ′
=− = =− =− = + =
−−
(2)
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.
#210340 - $15.00 USD
Received 16 Apr 2014; accepted 11 Jun 2014; published 18 Jun 2014
(C) 2014 OSA
30 June 2014 | Vol. 22, No. 13 | DOI:10.1364/OE.22.015571 | OPTICS EXPRESS 15572

We focus on the analysis of paraxial properties of a three-component zoom lens (Fig. 1),
where α,β,γ,, and δ are given by Eq. (1), 12 3
φ,φ,φ are values of the optical power of
individual components, and d1, d2 are distances between components of the zoom lens. We
can write for such zoom lens [24, 25] following formulas
21 2 121 11
α1(φφφφ)φ,ddd=− + − − (3)
12 212
βφ,dd dd=+− (4)
123 1212321312 12312
γφ(φφφ)φφ φ φ φφ ()φφφ ,dd dd dd=− =− + + + + + + − (5)
1 2 3 23 1223
δ1(φφ)φφφ.dddd=− + − + (6)
Consider an optical system presented in Fig. 1, where ξ is the object plane, ξ' is the image
plane and m is the transverse magnification of the optical system. The mutual distance
LAA
′
= of planes ξ and ξ' (Fig. 1) can be expressed as
12
LAA sd d s
′′
==−+++ (7)
and the distance DAF
′
= of points A and
F
′(image-space focal point) can be expressed as
12 .
F
DAF sdd s
′
′′
==−+++
(8)
Let us demand now that the distances LAA
′
= and DAF
′
= remain constant during the
change of magnification m. If we require the correction of field curvature of the third order of
the optical system [26–31], then it holds
123
0.62(φφ φ)0,
IV
S≈++= (9)
where
I
V
S is the Petzval sum.
Fig. 1. Three-element optical system.
By substitution of Eqs. (2)–(6) and Eq. (9) into Eqs. (7) and (8) we obtain after a tedious
derivation for the calculation of the distance 2
d the following nonlinear equation
432
42 32 22 12 0 0,cd cd cd cd c++++= (10)
where
#210340 - $15.00 USD
Received 16 Apr 2014; accepted 11 Jun 2014; published 18 Jun 2014
(C) 2014 OSA
30 June 2014 | Vol. 22, No. 13 | DOI:10.1364/OE.22.015571 | OPTICS EXPRESS 15573

23 3
41212 1 212
22 2
22
12 1 2
3 2 12 1 2 12 1 2
22
12 2 1 2
2
332 22
12 1 2
22121121211
1
,
φφ(φφ)[φφ(φφ)],
φφ(φφ)[( (4 φφφ)(2φ2φφφ))(φφ)
2φφ 4φ(φφ)],
φφ(φφ)[φ(φφ)φφφ(φφ)(3φ4φ(7 3))],
QLD
cQmQ
Q
cQmmD
m
mmQ
cmmQQmDmQm
m
c
=−
=+ ++
+
=+−−++
+− +
+
=++++−+−
=3
32 2 2
12 1 2
111
6
42 2
1
01
φφ (φφ)[2 3φ(2 1) φ(3φ2)],
φ
φ[(21)].
mmQmQD
m
Q
cm mDmQ
m
+
−+−+−
=+ +−
The distance 1
d can be calculated from the following equations
2
21 11 0 0,ad ad a++= (11)
where
2
212212 1
22
11 112 22222
2
02212
φφ (φφ)φ,
φφ(φφ)(2 φφ),
()(φφ)1/ 2,
ad
aL ddLd
amdLd m
=+−
=− + − +
=+ − + + −
and
2
21 11 0 0,bd bd b++= (12)
where
2
2 122 1 2 1
22
11 112 22222
2
02 212
φφ (φφ)φ,
φφ(φφ)(2 φφ),
()(φφ)1/ 2.
bd
bD dd Dd
bdDd m
=+−
=− + − +
=− ++−
The values L, D, 1
φ and 2
φ are given as input parameters. Other parameters can be
calculated by the following procedure. The distance 2
d is calculated using Eq. (10) for a
given value m of the transverse magnification. This distance is then substituted into Eq. (11)
and Eq. (12). The distance 1
d is calculated as the mutual solution of both Eqs. (11) and (12).
One has to choose the solution for the distance 1
d, which is the same for both equations. In
such case that Eq. (10) has complex roots or the value 2
d is negative, then we must change
the values of powers 1
φ and 2
φ in order the solution was real and positive ( 20d>).
In case that the image-space focal point
F
′ or the image point
A
′ lies inside the optical
system ( 0
F
s′
′<,0s′<), then one needs to put another optical system with the fixed focal
length behind the zoom system, which images points
F
′ and
A
′ as points
F
′′ and
A
′′ that
are located behind the system as one can see from Fig. 2. It is possible to place the spatial
filter in the plane, which pass through the point
F
′′ , and the image will lie in the plane, which
passes through the point
A
′′ . The position of points
A
′′ and
F
′′ is fixed and does not change
during the change of magnification.
#210340 - $15.00 USD
Received 16 Apr 2014; accepted 11 Jun 2014; published 18 Jun 2014
(C) 2014 OSA
30 June 2014 | Vol. 22, No. 13 | DOI:10.1364/OE.22.015571 | OPTICS EXPRESS 15574

Fig. 2. Combination of proposed zoom lens and fix focus lens.
3. Examples
We present an example of the calculation of basic parameters of the three-component zoom
lens with the fixed distance between object and image points and the fixed position of the
image-space focal point. We choose L = 150 mm and D = 110 mm. The results of the
calculation for different values of focal length 1
f
′ and 2
f
′ and different values m of the
transverse magnification are presented for three variants of the zoom lens in Tables 1, 2, and
3. All linear dimensions in tables are given in millimeters.
Table 1. Parameters of the First Variant of Zoom Lens (L = 150 mm, D = 110 mm)
125f′=, 230f′=, 313.6364f′=−
m
f'
d
1d2
s
s'
F
s
′
′
−5.00 8.0000 91.2808 5.9336 −31.6991 21.0864 −18.9136
−3.50 11.4286 85.2401 16.5327 −28.1472 20.0800 −19.9200
−2.50 16.0000 84.1556 25.3086 −22.6538 17.8820 −22.1180
−1.50 26.6667 90.9772 30.7193 −19.3833 8.9201 −31.0799
−1.00 40.0000 94.5709 32.6463 −25.1980 −2.4152 −42.4152
−0.75 53.3333 90.6227 36.2453 −31.3116 −8.1796 −48.1796
−0.50 80.0000 79.7162 47.6046 −32.2664 −9.5873 −49.5873
Table 2. Parameters of the Second Variant of Zoom Lens (L = 150 mm, D = 110 mm)
125f′=, 240f′=, 315.3846f′=−
m
f'
d
1d2
s
s'
F
s
′
′
−15.0 2.6667 8.0155 97.7797 −17.2516 26.9532 −13.0468
−12.0 3.3333 17.8174 88.5242 −16.6235 27.0350 −12.9650
−10.0 4.0000 25.4133 81.3555 −16.1797 27.0515 −12.9485
−8.0 5.0000 34.3673 72.8294 −15.8315 26.9717 −13.0283
−5.0 8.0000 53.6143 52.9911 −17.3779 26.0167 −13.9833
Table 3. Parameters of the Third Variant of Zoom Lens (L = 150 mm, D = 110 mm)
125f′=, 253f′=− , 347.3214f′=−
m
f'
d
1d2
s
s'
F
s
′
′
−6.0 6.6667 55.2590 46.3791 −35.8598 12.5021 −27.4979
−5.0 8.0000 37.9928 61.0085 −39.4650 11.5336 −28.4664
−4.0 10.0000 26.0162 67.7595 −44.2542 11.9701 −28.0299
−3.0 13.3333 16.7596 67.4990 −51.7488 13.9926 −26.0074
−2.0 20.0000 10.6715 53.3847 −65.6427 20.3011 −19.6989
#210340 - $15.00 USD
Received 16 Apr 2014; accepted 11 Jun 2014; published 18 Jun 2014
(C) 2014 OSA
30 June 2014 | Vol. 22, No. 13 | DOI:10.1364/OE.22.015571 | OPTICS EXPRESS 15575

4. Conclusion
A theoretical analysis of paraxial optical properties of the three-component zoom lens with
the fixed distance between object and image points and the fixed position of the image-space
focal point was presented in our work. Formulas for the calculation of paraxial parameters of
such zoom lenses were derived. The procedure of the calculation of parameters of zoom
lenses was presented on examples. Zoom lenses with the fixed distance between object and
image points and the fixed position of image-space focal point may find their applications in
optical systems for information processing, where it is possible to affect the amplitude and
phase using the spatial filter, which is positioned in the fixed focal plane of the zoom lens.
The advantage of the proposed solution, i.e. the zoom lens with the fixed distance between
object and image points and the fixed position of the image-space focal point, is the fact that
one can affect Fourier spectra of objects, which are characterized by the different spatial
structure, with just one spatial filter. The position of the image-space focal point does not
change with the change of focal length or the transverse magnification of the zoom lens and
the position of the investigated object stays also fixed.
Acknowledgment
This work has been supported by the Czech Science Foundation grant 13-31765S.
#210340 - $15.00 USD
Received 16 Apr 2014; accepted 11 Jun 2014; published 18 Jun 2014
(C) 2014 OSA
30 June 2014 | Vol. 22, No. 13 | DOI:10.1364/OE.22.015571 | OPTICS EXPRESS 15576