DESIGN OF ACHROMATIC AND APOCHROMATIC MICROOBJECTIVES
WITH PLASTIC LENSES
Grigoriy I. Greisukh, Evgeniy G. Ezhov, Il’ya X Levin and Sergei A.
Stepanov
Penza State University of Architecture and Construction, 28
Titov Street, 440028 Penza, Russia.
E-mail: grey@pguas.ru
This work shows the possibility and the efficiency of a single diffractive lens usage to achromatize and apochromatize microobjectives with plastic lenses. It also gives recommendations on assembling the source schemes of objectives and calculating the construction parameters required for a following optimization. It was also shown that achievable optical characteristics of achromatic and apochromatic microobjectives with plastic lenses satisfy the qualifying standards for cell-phone objectives and CCTV-cameras.
Introduction
As is well known, the first-order chromatism correction of refractive lens optical systems within the limited choice of optical materials can be achieved by means of diffractive lenses (DL) (see [1-5], for example). One of the possible ways to achromatize or apochromatize an optical system with the chromatism above the required level is shown in [6]. It implies the usage of diffractive-refractive correction unit, consisting of DL and one or two refractive lenses (RL). The correction unit can be build using both additional lenses and own RLs of the optical system, located near the aperture stop.
The present work gives recommendations for rational assembling the microobjectives scheme that allow to achieve achromatization or even apochromatization by means of merely a single additional DL. These

recommendations also open the way for further optimization basing on the calculated construction parameters. Both the recommendations and the calculations are given for photo- and cell-phone microobjectives and CCTV-systems of security cameras working in Day/Night vision, that is in the spectrum interval that includes visible and near infrared ranges from
 min

0,4 μm до
 max

0,9 μm.
The low cost requirement and decent optical characteristics of the given microobjective class stipulate the usage of optical plastic materials to produce the elements of their optical schemes. The modern shaping methods based on the precision punching allow replicating acrylic lenses with aspheric refractors and also allow applying a diffractive microrelief on demand [7,8].
Scheme assembling and achromat optimization results
The analysis of different vendors’ microobjective series of a given class [9] gave the following default values of their basic optical characteristics: the back focal length f
 
3,7 mm; relative aperture – 1:2,4; field angle in object space –
2

=60
○
. Such characteristics justify the starting scheme of an achromatic lens based on a triplet having its aspheric lenses made of the same crown-like plastic.
(
In the visible range, i.e. between blue F- and red C- spectrum lines of hydrogen
 min
 
F

0,48613 μm и
 max
 
C

0,65626 μm), it is possible to confine with achromatization that provides the equality of system image distances at the edges of the selected spectrum range ( s

 min
 s

 max
), due to a small width of a secondary spectrum. In a wider spectrum range that includes visible and near infrared ranges, the full exclusion of a primary chromatism influence to an objective’s resolution is only possible via apochromatization. It should provide the equality of system image distances on three wavelengths
 min
    max
, that is: s

 min
 s


 s

 max
.
Refractive indexes and Abbe numbers as well as all other optical parameters in this work are presented for the yellow d-line of helium (
 d

0,58756 μm) that was used for apochromatization alongside with
 min
=0,4 μm и
 max
=0,9 μm due to rigid requirements to the camera’s resolution in the Day vision; that is    d
.
As a crown-like optical material we have chosen polymethylmethacrylate (acrylic

PMMA) ( n d

1,491756;
 d

57,4408), and as a flint-like – polycarbonate (PC)
( n d

1,585470;
 d

29,9092).
To achromatize the triplet with all RLs build of PMMA it is all-sufficient to include a single DL into the scheme [10]. Figure 1 shows the optical scheme of such 4-lens achromat and Table 1 lists its construction parameters gained via the optimization.
Figure 1. Optical scheme of a 4-lens achromat: 1– aperture stop; 2 – DL.
Table 1. Lens Listing for Four - Lens Achromat
Surfa- ce
Radius
(mm)
OBJ: Infinity
1
2 b
1.7319
3.9729
STO: Infinity
4
5
-1.6791
-1.3310
Thickness
(mm)
Glass  
10
2
2 mm
-3
,
Infinity Air
0.7600 PMMA
0.1000
0.3425
Air
Air
-2.2695
-6.1431
0.6365 PMMA 2.1514
0.0500 Air 5.3303
Aspheric coefficients

3

10 mm
-5
2
,

4

10
2 mm
-7
,
-3.5615
-16.0129
-21.8536
3.6462
0.2676
-3.5652
17.5853
4.1340

5

10
2 mm
-9
,
-2.7021
8.5846
10.6459
6.7561

6
7
2.0066
1.5455
0.7000 PММА -14.6056
2.0556 Air -18.7189
8.4726
5.7059
-2.5418
-1.7250
0.4269
0.0791
IMG: Infinity 0.0000 a
3.712-mm FL b
Diffraction Order and Diffractive lens phase coefficients:
A
2

43.9432 mm
-4
; A
3

-493.7187 mm
-6
;
A
5

-7311.6339 mm
-10
;
A
8

551.7575 mm
-16
.
A
6

8711.5723 mm
-12
; m =1;
A
4

2909.2307 mm
A
1

-8
;
-107 mm
A
7

-4461.2733 mm
-14
;
-2
;
In Table 1 and further down
 i
, m и A j

are parameters of aspheric surfaces and DL’s ring microstructure applied to the back side of the aspheric surface of the first RL. The above parameters appear in equations that are used to describe optical elements, in particular, in optical design program ZEMAX [11].
Even aspheric surfaces are described with the following equation: z

1
 c

2
1
 c
2  2
 i


2
 i

2 i
, (1)
Where z is the coordinate of a surface point that is away from the optic axis to the length of

in the coordinates, bound to the vertex of this surface, ñ

1 r is a surface curvature in that vertex,
 i
where i

1 , 2 , 3 , ...
are aspheric coefficients.
The equation of the DL structure spatial frequency is:

(

)

2

1 m d
 d

, (2) where surface adds phase to the ray
  m j


1
A j

2 j
, (3)

is the distance measured from the optic axis, m
is the number of the diffraction order.
The DL with the structure described with (2) and (3) has the optical power determined by A
1
and m :
  
A
1
 m

. (4)

A j
, where j

2 , 3 , ...
, determine the contribution of the DL to the spherical aberration of 3 th , 5 th and subsequent orders.
The analysis of aberration curves of a microobjective with the parameters shown in Table 1 has shown that it is possible to combine achromatization with the significant reduction of the secondary spectrum by limiting the back focal length in all visible spectrum range with the value
 s
F
 
8 μm. It is also possible to significantly lower monochromatic aberrations and the spherochromatism. As a result, the polychromatic resolution of an objective is 98 mm -1 across the field angle 2   60 o
with the contrast K =0,5; and with K =0,78 it never lowers below
50 mm -1 . The distortion within the entire field of vision does not exceed

0,5%.
However this objective has a notable drawback. It is the quite wide range of field angles in the image domain that leads to a heterogeneous lighting of the multielement photodetector surface. Indeed, the main rays fall onto peripheral elements of the photodetector under the angle
  max

1, 12
 max
, which is 33,6º.
The above drawback can be mostly eliminated by including an additional RL as a terminal lens. Figure 2 shows the optical scheme of the five-lens achromat, and Table 2 lists its construction parameters gained via the optimization.

Figure 2. Principal optical scheme of the five-lens achromat: 1– aperture stop; 2 –
DL.
Table 2. Lens Listing for Five-Lens Achromat
Surfa- ce
Radius
(mm)
6
7
8
9
4
5
OBJ: Infinity
1
2 b
1.4784
8.5921
STO: Infinity
-0.9252
-1.7190
1.5861
2.0691
2.1243
2.1776
Thickness
(mm)
Glass  
10 2
2 mm -3
,
Infinity Air
0.6805 PMMA
0.1000
0.8018
Air
Air
-0.3036
-2.5530
0.3805 PMMA 22.1007
0.0500 Air -6.3740
0.5560 PММА -16.6813
0.2697 Air -0.2514
0.7675 PММА -24.8607
0.8068 Air -18.1545
Aspheric coefficients

3

10 mm -5
2 ,

4

10 2 mm -7
,
-5.0378
-2.9428
-30.3701
-9.7754
2.2752
-5.4875
11.2821
2.3803
IMG: Infinity 0.0000 a
3.7-mm FL b
Diffraction Order and Diffractive lens phase coefficients:
A
2

103.9467 mm
-4
;
A
5

-11266.4930 mm
A
3

-1054.0420 mm
-10
;
A
8

3313.1630 mm -16 .
A
6

-6
;
14704.7480 mm
-12
; m =1; A
1
-107 mm
-2
;
A
4

4797.3402 mm

-8
;
A
7

-10400.9350 mm
-14
;
6.9373
-7.0473
63.5122
31.1070
-2.1471
2.1004
-2.3697
-0.4134
Ray and wave spherical aberration curves and a polychromatic modulation

5

10 2 mm -9
,
-7.3209
3.2878
-9.6107
-9.6795
0.4165
-0.2941
0.1701
0.0686 transfer function curve for this microobjective are shown at Figure 3-6. The MTF curve has been calculated in a paraxial image plane. As follows from the curves, the resulting distorting plane-achromat with the relative aperture 1:2,4 provides the resolution of 165 mm -1 across the field angle 2
  60º with the contrast no lower than 0,5, and 100 mm -1 with the contrast of 0,69. The residual chromatism in the range from
 min

0,48613 μm to
 max

0,65626 μm does not exceed 6,61 μm, and the distortion is lower than 1 %. Main rays fall onto the peripheral elements of the photodetector under the angle
  max

0, 66
 max
, which is 19,8º.

Scheme assembling and apochromat optimization results
The studies have shown that apochromatization is achievable with an additional DL included into the optical scheme of a microobjective; however the scheme should satisfy several conditions.
First, alongside with positive lenses made of PMMA it should include at least one negative lens made of PC that would be the place for a DL.
Second, the source optical scheme should provide a way to limit the focal length drop within the selected spectrum range to the value equal to the spread of the main intensity diffractive distribution maximum in the focal region along the optic axis [12], even before the inclusion of a DL:
 f
  f

(
 max
)
 f

(
 min
)

16
 d
K
2
, (1) where K

is the (F-number). And the absolute value of Petzval surface’s radius, that an apochromatic optical system could form a stigmatic image [12] on, should satisfy the following condition (considering 2

=60
○
):
R

2 , 5 f

. (2)
The above requirements alongside with the need to limit coma and astigmatism could be fulfilled with a simple triplet, considering the acceptable surface curvature from the point of ray intersection heights with the relative aperture and field angle chosen above. In particular, this is stipulated by the fact that the starting optical scheme doesn’t need the spherical aberration correction since it initially supposes the inclusion of a DL and aspherizing of a refractor. However the residual spherochromatism of the achromat based on a triplet notably limits the resolution of a microobjective due to the extended spectrum range and enforces the usage of a scheme with four RLs, each one with two aspherical surfaces. The starting values of scheme’s construction parameters (refractor radiuses, lens thickness and air gaps, and also parameters for even aspherical surfaces) can be found using any known method of dimension or aberration calculation). However the authors have used the pseudo-ray method that implies a ray tracing approximation with different infinitesimal orders [13].

After placing a DL on the front surface of the second RL and the further optimization, the final optical scheme and construction parameters of the 5-lens apochromatic microobjective have been got (see Figure 7).
Figure 7. The optical scheme of the five-lens apochromat: 1– aperture stop; 2 – DL
A low level of monochromatic aberrations, a strict apochromatization with a small tertiary spectrum, a virtually zero distortion and a wide spectrum range polychromatic resolution not yielding to a 4-lens achromat resolution – these are the main advantages of a 5-lens apochromat. Unfortunately, its main rays fall onto peripheral elements under unacceptably big angles (
  max

1, 13
 max

34º).
It was possible to save the fulfillment of the above requirements and reduce the limiting values ratio of field angles in the image and object domains at more than
2,8 times (up to
  max

0 ,4
 max
) and nearly made it telocentric (
 max
 12º) by changing three RLs of a microobjective with two-lens glued elements. At the same time, the DL was moved onto the back refractor surface of the first glued element.
The optimization of this 8-lens scheme with only four aspherical surfaces has allowed to build a plane-apochromat with the focal length f

=3,7 mm that provides a resolution of 100 mm -1 with the relative aperture 1:2,4 and the contrast no less than 0,5 within the field angle range 2
  60º. The residual chromatism does not exceed 6,9 μm in the range from
 min

0,4 μm to
 max

0,9 μm, the transverse chromatism is comparable with the Airy disk and the distortion is less

than 1 %. The optical scheme and the construction parameters of the eight-lens plane-apochromat are shown at Figure 6 and Table 3 respectively.
Figure 6. The optical scheme of the eight-lens apochromat: 1– aperture stop; 2 –
DL.
Table 3. Lens Listing for Eight -Lens Apochromat
Surfa- ce
Radius
(mm)
Thickness
(mm)
Glass  
10 2
2 mm -3
,
Aspheric coefficients

3

10 mm -5
2 ,

4

10 2 mm -7
,

5

10 2 mm -9
,
OBJ: Infinity
1.3380 1
2 1.0817
Infinity
0.5169
1.0762
Air
PC
Air

STO: Infinity
4 -1.5219
5
6 b
7
-2.4540
-1.8295
3.4708
0.2738
0.9130
1.6574
Air
PC
0.3000 PMMA
0.1500 Air
PMMA
8
9
-4.7064 0.3500
-15.3523 0.1000
10 14.6809
11 2.5013
0.3500
PC
Air
PC
2.6787 PMMA
2.7333
3.9911
2.9526
7.1613
0.4239
-0.6118
-10.6113
0.0446
0.0809
12 -3.1078 2.1748 Air 3.0736 0.3135 -0.0586 0.0045
IMG: Infinity 0.0000 a 3.7-mm FL b
Diffraction Order and Diffractive lens phase coefficients:
A
A
2
5


-13.1132 mm
-0.6539 mm
-4
;
-10 ;
A
A
6
3


8.9955 mm
1.6548 mm
-6
;
-12 . m =1;
A
4

-7.7903 mm
-8
;
A
1

-11.8913 mm
-2
;
Ray and wave spherical aberration curves and a polychromatic modulation transfer function curve for this microobjective, calculated in a paraxial image plane, are shown at Figure 3-6. These curves clearly demonstrate the capabilities of the eight-lens plane-apochromat.
Notice that the DL structures of all microobjectives presented in this work are low-frequency and do not contain more than 10-15 ring zones. The low-frequency of structures allows manufacturing them with a notched stroke profile or even twoply, which is possible with the same optical plastic – PMMA and PC. This provides the nearly 100% effectiveness of a diffraction within the entire spectrum range [10].
9.0267
-0.0158
-0.0066

Conclusion
Summarizing the results of the present work, we can make the following conclusions:
− An achromatization in the visible range with the low level of a secondary spectrum is possible via the inclusion of a single DL to a microobjective having all its RLs made of the same crown-like plastic;
− The inclusion of a single DL to a microobjective having its RLs made of two marks of crown- or flint-like plastic allows an apochromatization in the wide range of wavelengths, including the visual and the near IR ranges;
− Authors give recommendations that allow assembling the source schemes of objectives and to calculate the construction parameters required for a following optimization;
− The present work has shown that achievable optical characteristics of achromatic and apochromatic microobjectives with plastic lenses satisfy the qualifying standards for cell-phone objectives and CCTV-cameras.
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