Design of High-resolution Fourier Transform Lens ZHANG Leia,b, ZHONG Xinga,b, JIN Guangb

（a Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences,

Changchun, Jilin, 130033，China b Graduate School of Chinese Academy of Sciences, Beijing ,100039）

ABSTRACT

With the development of optical information processing, high-resolution Fourier transform lens has often been used in holographic data storage system, spatial filtering and observation of particles. This paper studies the optical design method of high-resolution Fourier transform optical lenses system, which could be used in particles observation and holographic data storage system. According to Fourier transform relation between object and its frequency plane and the theory of geometrical optics, the system with working wavelength 532nm and resolution 3μ m was designed based on ZEMAX. A multi-configuration method was adopted to optimize the system’s lenses. In the optical system, a diaphragm was placed at the system’s spectrum plane and the system demanded a low vacuum to cut down the influences of atmosphere and other particles. The result of finite element analysis indicated that the influences of vacuum pumping to optics spacing and mirror surface shape very minor, and the imaging quality not being affected. This system has many advantages, such as simple structure, good image quality and a high resolution of 3μ m. So it has a wide application prospect and can be used both in holographic data storage system and particles observation.

Keywords: 4F optical lenses system, design, ZEMAX, analyze

INTRODUCTION

Fourier transformation is one of fundamental approaches to accomplish the optical information processing, and optical lens can complete the Fourier transformation of Two-dimensional images. Optical processing can perform a myriad of processing operations. This is primarily due to its complex amplitude processing capability. Optical Fourier transform processing is frequently used in practice, such as holographic data storage system, space wave filtering, and observation of particles. With the development of optical information processing, the need of high-resolution Fourier transform lens has been more urgent. For example, transforming images with it can test fields which can't be measured directly, like flow field, particles field, etc. So, the study on design and analysis methods of high-resolution Fourier transform lens system has very important meaning. [1]

Early lens design and analysis is a laborious work for the designers. The lens designers start with drawing light rays originating from an object. These rays pass through a glass surface and are bent or refracted through to the image plane. That may take a few months of work to design a simple system. Today, the using of computer and affordable optical design software makes the process of designing lenses more direct and much faster. This paper discusses the optical design and analysis method of high-resolution Fourier transform optical lens, which would be used in particles observation and holographic data storage system. By using the software of ZEMAX and UG, the system with a working wavelength of 532nm, a resolution of 3µm was designed and analyzed.

CHARACTERISTICS OF HIGH-RESOLUTION FOURIER TRANSFORM LENS

The layout of high-resolution Fourier transform lens in common use is shown in Fig.1. In the figure, L1 and L2 are the first and the second Fourier transform lenses. P1 is the system’s input plane. P is the system’s Fourier spectrum plane. P2

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is the system’s output plane. According to the Abbe image principle, the Fourier lens transforms the information on the input plane, and the swing distributing on plane P3 is two times diffraction of swing distributing on plane P1. The relation between image plane coordinate ( )yx ′′, and object plane coordinate ( )00, yx is:

⎪⎪⎩

⎪⎪⎨

⎧

=′

=′

01

2

01

2

yffy

xffx

(1)

where 1f and 2f are front and back focal length of the Fourier transform lens.

Fig.1 Fourier transform lens

Through comparing with other optical systems, high-resolution Fourier transform optical lens has many characteristics. [2-4]

1) A single lens cannot be used to perform a Fourier transform when the field angles are not small or the chief rays are not close enough to the optical axis. Because the nonlinear terms of the field angle may not be neglected, and the system can’t be capable of transforming plane waves into perfect diffraction limited point images and point objects into plane waves at this time. In order to achieve an accurate Fourier transform, a lens system of more than one component is often used.

2) The aberrations at two conjugated positions should be controlled. The first one is when the object is at infinity, the aperture stop is at the front focal plane and the image is at the back focal plane. The second one is when the object is at the front focal plane, the aperture stop is at the back focal plane and the image is at infinity. So to control aberrations of two conjugated positions, aberrations on both object plane and pupil should be controlled.

3) If accurate Fourier transform is required in the whole field, Fourier transform lenses must be designed according to the principle of diffraction imaging. The Fourier transform lens should fulfill sine condition.

4) According to Eq. (1), the amplification systems can be designed. By using this kind of system with same resolution, details of objects at the input plane can be well observed.

5) Generally, the high-resolution Fourier transform lens’s field of view is very small. At this time, the system may need many mirrors and changing the material of a few mirrors.

DESIGN OF THE HIGH-RESOLUTION FOURIER TRANSFORM LENS

3.1 Main parameters of the lens

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There are four main parameters in high-resolution Fourier transform lens. They are working wavelength, the cut-off spatial frequency maxN , the diameter of input plane 1D and the diameter of output plane 2D .The system should also satisfy the requirements of the dimension, the working distance and the imaging quality. To make sure that all the first-order diffraction can pass through the lenses, the cut-off spatial frequency maxN , the front focal length 1f , the diameter of spectrum plane fD should have the relationship: [5]

1max 2 fDN f ⋅= λ （2）

According to the theory of geometrical optics, the diameter of input plane 1D and output plane 2D , the front focal length 1f and the back focal length 2f should have the relationship:

2121 ffDD = (3)

Spatial-Bandwidth Product is defined as：

11max1 2 fDDND f ⋅⋅=⋅ λ (4)

Spatial-Bandwidth Product is equivalent to the Lagrange-Helmholtz invariant in geometrical optics. It denotes the amount of information that the system can transfer. According to Eqs. (2) to (4), the higher resolution that the system needs, the larger diameter of spectrum plane ( fD ) and the shorter front focal length ( 1f ) should be used.

3.2 Aberrations Correction

No chromatic aberration correction is necessary since the system uses monochromatic light with a wavelength of 532nm. Other aberrations should be all corrected for a Fourier transform lens. Telecentric optical path structure in the object space is used in the design, that is to say, object is at front focal plane, aperture stop is at back focal plane, and entrance pupil is at infinity. So, if the aberrations of the first conjugated position are well corrected, the aberrations 1S 、 2S 、 3S

、 4S of the image at the spectrum plane are also corrected. And if the spherical aberration and coma at the spectrum plane are well corrected, the spherical aberration and coma at the entrance pupil are also corrected. The lenses are placed symmetrically with respect to the spectrum plane, and the distortion can be well balanced. To correct the spherical aberration, positive-lenses and negative lenses can be combined. [5, 6]

3.3 Application of Multi-configuration Optimization Method

It is known through the above analysis that the design of Fourier transform lens is similar to the design of a special telescope. The telescope can be used at both positive and negative positions and its aperture diaphragm is at the object focal plane. So, optimizing the positive and reverse optical path respectively can correct all aberrations of Fourier transform lens. A multi-configuration model is established to optimize the lenses in ZEMAX. In the model, the optical parameters, such as field of view, and optimization merit function are set respectively, and the lens parameters and optics spacing in configuration 2 are picked up from configuration 1. The process is as follows: first, modeling multi-configuration and optimizing the front Fourier transform lens; then, scaling the front lens to produce the back Fourier transform lens and optimizing it; finally, combining the optimized front lens and back lens to obtain the required high-resolution Fourier transform lens system. 3.4 Design Requirements and Results

In this paper, a system which can be used in a particles observation test is discussed. The experiment needs a Fourier transform lens with a working wavelength of 532nm, a object distance of 160 mm～200mm, a field ofφ 20mm and a resolution higher than 3µm. So the system’s numerical aperture:

088.00.5 ≥= σλNA (5)

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HLons Data Editor

&d!t 1: w0t: OdPSf:T9o Caao!t Radllo Thik!o 9&—Dioto

032 Sto!dod I!fi!Uty -353.888000

-35.000000 0.000000

—i Sto!dod —2502.000000 15.700000 LM0Z 43. 976154 0.000000

Sto!dod -539.790000 18.500000 43.498837 0.000000

Th Sto!dod —135.935000

-10.500000 ZF2

-43. 573301 0.000000

—

Th Sto!dod 5000-000000-

23-750000-

46. 043107 0-000000—

S Stdod —3739-000000 23-2S0000 L72 51. 004016 0-000000

4 Stdd —173-SS0000-

1.700000-

63.321933 0-000000—

7 Stdd 374.460000 21.100000 L2 64.320389 0.000000

8 Sto!dod -1300.000000 68.800000 63.030638 0.000000

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To observe details of the particles at the input plane, amplification lenses system should be designed. A magnitude ratio of 2 is set in the design. So,

12 2 ff = (6)

In this paper, we select 1f =211 mm, then from Eq. (6), 4222 =f mm. A coaxial high-resolution Fourier transform lens system has been designed, and the whole process was optimized using ZEMAX with the wavefront error as the merit function. In the design, a diaphragm was placed at the system’s spectrum face and the system demanded a low vacuum to cut down the influences of atmosphere and other particles. Some optical parameters of the lens are listed in Fig.2.

Fig.2 Lens Data layout

ANALYSIS OF THE LENS SYSTEM

Many types of analysis in ZEMAX indicate that all aberrations were well corrected, and the results of these analyses also had guided the lens’s optimization. In this section, the final analysis results in ZEMAX are given, and the influences of vacuum to optics spacing and mirror surface shape are analyzed. [7]

4.1 Analysis Results in ZEMAX

The modulation transfer function curves and aberration curves at different field angles are shown in Fig.3 and 4 respectively. The field curves and distortion of the system are shown in Fig.5. The figures indicate that the lens has little image distortion and curvature, narrow longitudinal aberration(less than 0.01mm) and a wavefront error of 0.25λ . These results prove that the high-resolution Fourier Transform Lens can satisfy the requirements of the particles observation test.

Fig.3 MTF curves Fig.4 Aberration curves

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•_mrn3. So5orl I. SUBCSE - STATIC L0DS ID!SpI ocement - Nodal, UoQn!tuO,u!n: 0.000..000, Max: T.056.-003 mmDerormot!on S DspI ocemenS - Nodal

U l.008e003

U l.003e0039.984, -004

9.784e-O049.734.-004

9.684,-0049.633e-004

U 9.583e0049.633. -004

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•_mrn3. Sal So I. SUBCSE - STATIC L0DS ID!SpI acement - Nodal, UoQn!tuO,u!n: 0.000,.000, Max: T.056.-003 mmDerormot!on S DspI ocemenS - Nodal

U 5.245e004

U 5.206e004— 5.167e0045. 128e004

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5.01 le-004

4 .972e-004

4 .933e-004

4.894e-004

U 4.855e0044.818. -004

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Fig.5 Field curves and distortion of the system

4.2 Influences of vacuum

Keeping vacuum at the spectrum face can reduce influences of environment. Because vacuum pumping can clean out the dust motes and diminished the disturbance of atmosphere at the spectrum face. But refractive index and optics spacing are changed at the same time. Increasing drawtube thickness can reduce the optics spacing changes caused by the atmospheric pressure. Taking 6mm as the thickness of the drawtube and aluminium as material, aberrations were analyzed using finite element analysis software. The results indicated that optics spacing diminished less than 1.3µm and the biggest PV of the two lenses at the two ends of the vacuum chamber is 29nm. Distortions of the two lenses are shown in Fig.6. So, distortions are in the allowable range of the Fourier Transform Lens. The analysis results in ZEMAX indicate that the imaging quality was not affected.

Fig.6 Distortions of two lenses

CONCLUSION

In this paper, the design of high-resolution Fourier transform lens is discussed in detail. A Fourier transform lens with a working wavelength of 532nm, a field ofφ 20mm and a resolution of 3µm was designed using ZEMAX for a particles observation test. Analysis results in ZEMAX and UG indicated that the performance of the system can satisfy the requirements of the particles observation test.

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VHDSS”. Optical Technique, VOL.30 NO1, 2004. 5. WANG Zhijiang, Handbook of Optics Technology, Press of mechanical industry, Beijing, 1991. 6. AN Baoqing, XIANG Libin, GAO Zhan, XUE Mingqiu, “Study on precise symmetric Fourier Transform Lens”.

ACTA Photonica Sinica, VOL.26 No10, 1997. 7. ZEMAX optical design program users guide, ZEMAX Development Corporation, 2004.

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