MTF of compound eye Hamid Reza Fallah and Ayatollah Karimzadeh* Physics Group, University of Isfahan, Iran *
[email protected]
Abstract: Compound eye is a new field of research about miniaturizing imaging systems. We for the first time introduce a dual compound eye that contains three micro lens arrays with aspheric surfaces. The designed dual compound eye in one state is a superposition system in which each channel images all of field of view of the system. With adding a field stop we have decreased the stray light. MTF of ideal superposition compound eye calculated. Also with changing field stop the system is converted to an apposition compound eye in which each channel images only a part of total field of view and so the field of view is larger than that of superposition type. ©2010 Optical Society of America OCIS codes: (110.0110) Imaging systems; (350.3950) Micro-optics; (040.1240) Arrays.
References and links 1. 2. 3. 4. 5. 6.
R. Völkel, M. Eisner, and K. J. Weible, “Miniaturized imaging systems,” Microelectron. Eng. 67–68, 461–472 (2003). J. Duparré, P. Schreiber, and R. Völkel, “Theoretical analysis of an artificial superposition compound eye for application in ultra at digital image acquisition devices,” in Optical systems design, Proc. SPIE 5249, SPIE, (St. Etienne, France), September 2003. J. Duparré, P. Schreiber, A. Matthes, E. Pshenay-Severin, A. Bräuer, A. Tünnermann, R. Völkel, M. Eisner, and T. Scharf, “Microoptical telescope compound eye,” Opt. Express 13(3), 889–903 (2005). H. R. Fallah, and A. Karimzadeh, “Design and Simulation of a high resolution superposition compound eye,” J. Mod. Opt. 54(1), 67–76 (2007). N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A, Pure Appl. Opt. 4(4), 1–9 (2002). M. Gu, Advanced Optical Imaging Theory (Springer, 2000).
1. Introduction With recent improvements of image sensors and microlens arrays technology the interest in miniaturizing imaging systems such as compound eye systems has been increased [1,2]. There are two different types of compound eyes, the apposition and the superposition compound eye systems [3,4]. In this paper a dual compound eye system with two special field stops has been designed using paraxial calculation and optimization techniques. With dual system we mean a system which can work as both superposition and apposition types. This is done by designing and applying a suitable field stop. In Section 2 superposition and apposition compound eye systems are discussed and paraxial ray tracing of microlens array for superposition and apposition compound eye simulation is reported. In Section 4 the design and optimization of a superposition eye system with special field stop is reported. In Section 5 the MTF calculation of this system is presented and in Section 6 converting this system to apposition eye with designing a convenient field stop are described. Finally in Section 7 conclusions are presented. 2. Compound eye systems The designed compound eye consists of three micro lens arrays (Fig. 1). p1, pf and p2 are respectively pitch size of first, second and third microlens of each channel and their focal lengths are f1, ff, and f2. Also t is air space between the second and the third arrays and L is image distance from the third array.
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(C) 2010 OSA
Received 14 Oct 2009; revised 1 Jan 2010; accepted 17 Jan 2010; published 26 May 2010
7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12304
Fig. 1. Compound eye layout.
In apposition eye system each channel only transfers a portion of the overall field of view (FOV) and the adjacent channel’s FOVs are attached to each other. On the other hand in superposition compound eye each channel can transfer the full FOV of the system. The ability to concentrate light and numerical aperture (NA) of superposition compound eye system is very large compared to the apposition eye one. On the other hand, FOV of apposition eye is larger than that of superposition eye [3,4]. 3. Paraxial ray tracing The paraxial ray trace matrix for N-th channel from central channel after passing through three micro lenses is as follows according to the notation in Fig. 1 [4]:
- t f1 B = (t − f 2 ) f1 f 2 0
− f1
f1 + t (1 − f1 f f ) f 2 − (t − f 2 )(1 − f1 f f ) f 2 0
Ntx − Np1 N(f 2 − t ) x f 2 + Ny 1
in which (1) x = ( p1 - p f ) f f , y = ( p1 - p2 ) f 2 The rays must be focused at a distance L from third microlens array after exiting from channels. So this transformation matrix is: hout hin 1 θ = A out θ in = 0 1 1 0 1 L where A = 0 0 0 0
L 0 hin 0 0 B θ in 0 1 1 0 0 B 1
(2)
where hin and θin are height and slope of incident ray and hout and θ out are height and slope of the exit ray in image surface respectively [5]. To focus all rays with same angle and different height on each microlens A11 and A13 must be zero: p2 p2 t, L = t (3) p1 p1 - p2 The chief ray in each channel should pass through the center of third microlens to get low aberrations and so: A11 = 0, A13 = 0 ⇒ f 2 =
#117382 - $15.00 USD
(C) 2010 OSA
Received 14 Oct 2009; revised 1 Jan 2010; accepted 17 Jan 2010; published 26 May 2010
7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12305
( p p )t tf1 , p f = p1 - 1 2 (4) t + f1 t + f1 To hold each channel's FOV equal to the overall system’s FOV in superposition eye system we should have: B12 = 0, B13 = N ( p1 - p2 ) ⇒ f f =
pf FOV f1 + N max ( p1 - p f ) = 2 2 On the other hand, for apposition compound eye system we have:
(5)
pf FOV + N ( p1 − p f ) = (6) 2 2 Equation (5) and (6) determine maximum number of arrays (N) relation with FOV in superposition and apposition eye systems respectively. Using Eqs. (1) and (5) the image size, D, could be calculated as: f1 f 2 FOV t - f2 Equations (1–7) have been used for initial designing of compound eye system. D=
(7)
4. Design of a typical dual compound eye system
As an starting point, we consider a system with total length of 9 mm, aperture of 3mm × 3mm and image size of 1.26 mm. We choose f1 = 2mm , t = 2mm, p1 = 1mm and p2 = 0.75mm . Using Eq. (3) and (4)
f2 , f f , p f
and L are calculated as: f 2 = 1.5mm , f f = 1mm ,
p f = 0.875mm and L = 6mm. The calculated paraxial parameters are used as starting values for the layout of a system consisting of real array of microlens. With the use of convenient error function, aberrations especially astigmatism and coma have been corrected in each channel. Throughout the optimization process, the size of p1 , p f , p2 have been changed to 0.622mm, 0.747mm, and 0.872mm respectively. The constructional parameters of the designed system are shown in Table 1 and the layout of the three channels of the compound eye system along with the ray paths are shown in Fig. 2(a). FOV of each channel is 12° and the image size of final system is 1.1mm for superposition and 1.66 mm for apposition compound eye. The veiling glare can be reduced with designing and inserting a field stop at the field lens array). A designed field stop for this task is shown in Fig. 2(b). In this figure, the large circles are microlenses and smaller circles are those field stops which can be implanted on a plane. The central channel’s field aperture is a circle with 0.48 mm diameter and the other field stops have the same size but with a decenteration of 0.24 mm with respect to the center of each microlens. Diameter of each microlens in field lens array is 0.747mm.
Fig. 2. a) Layout of superposition compound eye, b) Field stop for decreasing veiling glares (Colored areas are field stops).
#117382 - $15.00 USD
(C) 2010 OSA
Received 14 Oct 2009; revised 1 Jan 2010; accepted 17 Jan 2010; published 26 May 2010
7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12306
. Maximum distortion of each optimized channel is less than 0.5% [4]. MTF of one channel for 50 cycles/mm is about 0.2 (Fig. 3). Table 1. One channel surfaces specifications SURFACE OBJECT
RADIUS PLANE
THICKNESS 0
APERTURE RADIUS 0.07
GLASS AIR
CC
AST 2 3
2 −2 1.01
0.2 1.965584 0.2
0.436 0.436 0.3735
K11 AIR K11
−37.938798 −1.867479
4
−1
1.965584
0.3735
AIR
5 6 IMAGE
1.5 −1.5 PLANE
0.15 5.141116 0
0.311 0.311 0.57
K11 AIR
5.184085 −2.139013
Fig. 3. MTF of one channel of superposition compound eye.
5. MTF calculation of the superposition eye system
In superposition compound eye system there are overlapping between images of different channels which should be considered in MTF calculation. We calculate MTF for the aberration free superposition compound eye case to find diffraction limited criteria. For MTF calculation we have to calculate autocorrelation of the pupil function [6]
MTF (α , β ) =
∫∫ p( x +
α fλ 2
,y+
β fλ 2
) p* ( x −
α fλ
∫∫ p( x, y )dxdy
2
,y−
β fλ 2
)dxdy (8)
Where f is focal length of eye The pupil consists of 9 circles with radius r. For simplifies we calculate MTF along x axis as shown in Fig. 4.
#117382 - $15.00 USD
(C) 2010 OSA
Received 14 Oct 2009; revised 1 Jan 2010; accepted 17 Jan 2010; published 26 May 2010
7 June 2010 / Vol. 18, No. 12 / OPTICS EXPRESS 12307
a)
Fig. 4.
For
λα f
λα f 2r
< 1, b)1
