Design of paraxial diffractive elements with the CAD system DIGIOPT Harald Aagedal, Thomas Beth, Heiko Schwarzer, Stephan Teiwes Institute of Algorithms and Cognitive Systems, University of Karlsruhe Am Fasanengarten 5, D–76128 Karlsruhe, Germany Tel.: +49-721-6084213, Fax: +49-721-696893 E-mail:
[email protected]
Abstract User-friendly computer-aided design tools for the implementation of diffractive optical elements may on the one hand influence the acceptance of diffractive optics in industry and on the other help to simplify research work in this area. In the present paper we present the software system DIGIOPT which was designed to fulfil our demands on the design of paraxial diffractive elements for optical information processing applications. Key words: DIGIOPT, DE design and simulation, optical information processing.
1. Introduction Diffractive optics is a field of growing importance not only in science but also for industrial applications such as information and material processing. In order to make diffractive optics more attractive to industry easy-to-use design and simulation tools with a high amount of functionality and flexibility have to be developed. An important aspect of diffractive optics is the modelling of inverse wave propagation under consideration of certain boundary conditions such as optical function, material constraints and structure size. On the basis of Fourier optics, methods to design paraxial diffractive elements for applications in information processing, communication technology, laser physics, etc., are developed. In this context, the computer-aided design (CAD) of diffractive elements has several advantages compared to optically produced diffractive elements, • simple and flexible modelling of digital signal waves, • automated diffractive element design by efficient linear and non-linear optimisation techniques, • test by simulation of the optical function of the designed optical system and • direct implementation of digital elements, e.g. e-beam lithography or addressable spatial light modulators (SLMs) In principle, commercially available standard software can be used to design diffractive elements [1]. However, the required flexibility and functionality are difficult to achieve with standard software. The design and coding of diffractive elements such as filter, multifacet or transformation elements using standard software, e.g. Mathematica, require a great amount of programming which can only be done by experts knowing the recent design and coding techniques. At present, there are hardly any software packages combining developed theoretical concepts in a structured way under a comfortable user interface, thus making diffractive optics available to experts as well as non-experts of the field. There are many criteria that have to be fulfilled by a CAD system for the design and simulation of diffractive elements. These can be summarised into the following keywords: • Functionality. The most important feature of a software system is of course that it is capable of doing what the user wants it to do. In diffractive optics this means that a variety of standard low level operators as well as high level algorithms for the design of diffractive elements should be contained in the system. Physically correct simulation of diffractive elements and wave propagation should also be carried out by the system so that entire optical systems can be simulated correctly.
• Flexibility. Flexibility of a program for diffractive optics is the ability to combine all kinds of operators in every imaginable way. This is of great importance for the scientific utilisation of the program. Flexibility enables the user to test new ideas and perform rapid prototyping of diffractive elements and entire optical systems. • User-friendliness. User-friendliness should be a property of every end-user application. This criterion is however very hard to fulfil and normally only found in application software with a broad market such as word processing systems, spreadsheets, etc. This is unfortunately not the case of systems for the design of diffractive elements. • Communication. The software system should be capable of communicating with production devices such as e-beam machines, laser printers, etc.. For scientific documentation it should also be possible to acquire bitmaps, graphs, tables, etc. which can be pasted into standard text processing systems. A program that fulfils these criteria can be used as a design and simulation tool for a scientific and industrial purpose as well as in the education of non-experts of diffractive optics. Thus, such a program could function as a link between research and industry, making new theoretical concepts of diffractive elements available to industry and perhaps even further the acceptance of diffractive optics for industrial utilisation. In the present paper we will describe the software system DIGIOPT for the design and simulation of optical systems based on paraxial diffractive elements which is a good attempt in this respect.
2. The Concept of DIGIOPT The software system DIGIOPT was designed and implemented at the Institute of Algorithms and Cognitive Systems, University of Karlsruhe with the above criteria in mind. It runs under Microsoft Windows on IBM compatible PCs. DIGIOPT is strongly windows oriented. A wavefront or “signal” is represented by a complex-valued two dimensional array of which either the amplitude, phase, real or imaginary distribution is shown in a window on the DIGIOPT desktop (Figure 1). A variety of elementary “preprocessing” operations can be performed on each window representing the signal. These include • unitary operators for digital preprocessing of 2D signals such as scaling, translation, rotation, embedding, Fourier, Fresnel and Kirchhoff transformation, etc., • binary operators such as addition, subtraction, multiplication, division, convolution, correlation, etc. of signals, • functions for the generation of primitive wavefronts such as plane, spherical, cylindrical and gaussian waves, etc., as well as parsing of analytical expressions,
Figure 1 The windows-oriented concept of DIGIOPT. The amplitude, phase, real and imaginary distribution of a Fresnel transform of the upper left image are shown in the upper row. A complex histogram and a profile of the Fresnel transform are displayed in the lower row.
• functions for the computation of complex histograms and profiles of signal distributions and • functions determining the quality of diffractive elements such as diffraction efficiency and signal-to-noise ratio with respect to the generated signal wave.
One of the goals of DIGIOPT was to gather all relevant algorithms for the design of paraxial diffractive elements in one userfriendly program, including recent work concerning the iterative Fourier transform algorithm [2]. The “high level” features may be summarised into • coding operators, • a general form of the iterative Fourier transform algorithms, • operators for the generation of multifacet elements, • 2D display of wave propagation of one dimensional signals and • an animation facility (Figure 2). DIGIOPT also contains a simple programming language. This is an extremely helpful tool Figure 2 Animation tool whenever operators are to be concatenated or signals combined in some way or another. The “programmer” may also use loops as in the following program giving the animation sequence shown in Figure 2. var field_Object; var ani_a("TEST.ANI"); var real_d,card_Cycles;
// Creates an animation window on the DigiOpt desktop
field_Object:=EmbedExtractField(LoadField("marilyn.de"),128,128); // Loads and embeds the Marilyn image (64x64) // into a zero matrix of size 128x128 real_d:=15.0; card_Cycles:=50;
// Distance in mm // Number of iterations
repeat card_Cycles times real_d:=real_d*1.05; AddToAnimation(ani_a,FRT(field_Object,real_d)); // Adds Fresnel transforms of field_Object at distance // real_d to animation ani_a end;
The sequence is a collection of Fresnel transforms from distances between 15 mm and 172 mm of the image shown in Window 1 of Figure 5. As depicted in Figure 3, the DIGIOPT programming language may also be used to generate wavefronts defined by analytical expressions.
Figure 3 A one-line DIGIOPT “program” creating the distribution of Window 1. As mentioned in the introduction, a software system should be capable of communicating with production devices such as ebeam machines and laser printers. Through the MS Windows environment DIGIOPT can produce diffractive element distributions in a variety of data formats such as PostScript, GIF and BMP. At present, DIGIOPT does not support GDS2, but this will change in near future. For scientific documentation, bitmaps, graphs, tables, etc. can easily be exported from DIGIOPT and pasted into standard text processing or slide presentation systems. Likewise, online documention of research work as well as real-time simulation presentations with DIGIOPT running on a laptop connected to an overhead display are possible.
DIGIOPT was not only designed to simulate an optical bench, but also to be a component of opto-electronical hybrid systems (Figure 4) [3]. In such systems SLMs can be controlled by DIGIOPT. Moreover, the output of the optical system can be captured by a CCD camera and input into DIGIOPT. This makes it possible to perform real-time optical tests of diffractive elements [4].
3. Design Example
Projection lens SLM
Beam splitter
Input
Beam expander
CRT
Laser Fourierlens
DIGIOPT CCD
Output
Design of diffractive elements can be a complicated task. In order to give an impression on how a non-trivial design problem can easily be tackled by the DIGIOPT system we Figure 4 DIGIOPT as a controlling component of an optoelectronical hybrid system. design a non-periodic diffractive phase element generating a speckle-free reconstruction of a specified pattern in the Fourier plane. There are several steps which have to be performed in order to yield a diffractive element with satisfactory properties. We will try to focus on how to use DIGIOPT and not on the theoretical background. We have also chosen to use elementary operations which may be performed by higher level operations in one step so that the flexibility of DIGIOPT will become clear. Generating a non-periodic diffractive Fourier element, that is, a diffractive element with a continuous impulse response, is much more difficult than generating a periodic diffractive element with discrete impulse response. This is due to the sinc-interpolation of the digital impulse response which again is caused by the finite size of the diffractive element. For a discrete impulse response this sinc-interpolation does not do any harm, it simply changes the shape of the reconstruction points. In the nonperiodic case, however, speckles in the continuous impulse response will occur if the phase values of the reconstruction points are not carefully chosen. The interpolation problem with unfavourably chosen phase distributions can easily be demonstrated by DIGIOPT Figure 5 Speckles occur after sinc-interpolation if the phase values are not as shown in Figure 5. Window 2 was carefully chosen. generated by superimposing random phase values onto Window 1. To sinc-interpolate Window 2 we simply compute its Fourier transform (Window 3), embeds this into a zero matrix (Window 4) and compute the inverse Fourier transform (Window 5). This is the sinc-interpolation of the signal with the random phase, containing a great amount of speckles. We will now design a non-periodic diffractive phase element with a version of the iterative Fourier transform algorithm (IFTA). Iterative Fourier transform algorithms were first applied to phase retrieval problems in electron microscopy by Gercherg and Saxton [5]. These algorithms are based on an iterative technique which makes the transmission function of the diffractive element fulfil certain constraints in the Fourier plane as well as in the reconstruction plane by allowing some parameters of freedom in the reconstruction plane. In the case of intensity signals, these parameters are freedom of phase and scalation in a signal window in which the desired image is located and freedom of phase and amplitude outside this signal window [2].
3.1 Finding an object independent band-limited phase distribution The success of the iterative Fourier transform algorithms concerning the quality of the generated system responses depends on the initial distribution of either the diffractive element or of the signal, depending on whether the iteration starts in the Fourier domain or in the spatial domain. In our case, the iteration begins in the spatial domain. Hence, the initial phase distribution of the object has to be carefully chosen. On the one hand, the starting phase should function as a diffusor, that is, spread the energy into the entire spectrum of the complex object distribution. This can easily be done by superimposing a random phase distribution onto the object. On the other hand, the phase distribution has to be smooth to avoid the speckle effects shown in Figure 5. These initial speckles will not be removed by the iterative Fourier transform. In this sense, smoothness means that the phase values of two neighbouring pixels do not differ by values close to π. This is related to the existence of so-called 2π-jumps [6,7,8]. A random phase distribution is thus not a good choice. In our example we will use an algorithm proposed by Bräuer [7] to find an initial phase distribution. The Bräuer phase distribution was superimposed onto the plane wave in Window 2, giving Window 3, by selection the point “Bräuer Start Phase” of the dialog box in Figure 7. As shown in Window 4 of Figure 6, this “diffusor” spreads out the energy into the entire spectrum (Window 4). In order to achieve a smooth object-independent band-limited phase Window 3 has to be interpolated. This may be done by embedding Window 4 into a zero matrix (Window 5), computing the inverse Fourier transform (Window 6) and setting all amplitudes to one. Thus, the phase distribution of Window 8 is band-limited and evenly distributed in the central part of the spectrum.
Figure 6 The “Modify Phase” dialog (left) was used to find an initial phase distribution without 2π-jumps (Window 8).
3.2 Finding an initial object dependent band-limited phase distribution The object independent phase distribution found in Section 3.1 may now be superimposed onto the object. One should, however, be sure to have a band-limited object, that is, the spectrum should not exceed the “central one-quarter” of the spectral window. This may be achieved by simply sinc-interpolating the object as shown in Windows 1–5 of Figure 7. The pointwise multiplication of the object independent phase distribution and the interpolated object (Window 13) is now almost band-limited (Window 14). A simple iterative Fourier transform algorithm may now be applied to improve the phase distribution of Window 13 even further. This can be done by selecting the point “Band Limited Phase (IFTA)” of the “Modify Phase” dialog (Figure 6).
Figure 7 Before superimposing the object independent band-limited phase distribution onto the object, the object should be sinc-interpolated (Windows 1–5) to ensure band-limitation. The pointwise multiplication (Window 13) of Window 5 and 12 is now almost band-limited, as shown in Window 14. By applying a simple IFTA-version the phase distribution of Window 13 may be further improved (Window 15 and 16).
3.3 Generation of the diffractive element After having performed the steps in Sections 3.1 and 3.2 we have superimposed a phase distribution onto the image which the diffractive element is to generate as an intensity distribution in the Fourier plane. As already mentioned in Section 0, the iterative Fourier transform needs parameters of freedom in order to achieve diffractive elements with satisfactory properties. For intensity signals these are freedom of phase and scalation in a window containing the desired signal and freedom of phase and amplitude outside the window. The signal window may be selected by simply using the mouse or defining the coordinates (Window 21 in Figure 8). The size of the signal window is crucial for the quality of the diffractive element. If there is not enough space outside the signal window, the freedom of amplitude and phase will normally be too small to obtain an element with a satisfactory signal-to-noise-ratio [2]. Hence, the signal (Figure 8, Window 17) was embedded into a zero matrix (Window 21) and the signal window of Window 21 was selected to hold the signal with a margin of 4 pixels. The diffractive element designed was to be illuminated by a plane wave. Because of the speckle reduction we embedded a plane wave, i.e. a field I(x) with I(x)=1, into a zero matrix of the double size in order to simulate the finite size of the diffractive element. We could of course have chosen any other field (also a complex-valued one) as the illumination field of the diffractive element.
To start the iteration of the diffractive element the signal field (Window 21) and the illumination wave (Window 22) have to be specified in the “Diffractive Element Iteration” dialog box. If no illumination wave is specified, a plane wave (unembedded) is assumed. The type of the diffractive element may either be set to Amplitude, Phase or Complex, whereas
Figure 8 The dialog box for a general iterative Fourier transform algorithm. Window 21 selected as signal field, whereas Window 22 defines the illumination wave. the signal type is either Intensity or Complex. In our case we choose a diffractive phase element and an intensity signal. The “Animation” group makes it possible to save the result of every iteration in two animation sequences—one for the diffractive elements and one for their impulse responses. This lets the user examine the development of the diffractive element during the iteration process. The “Freedom iterations” group (Figure 8) should be described more carefully. In this group the number of iteration cycles has to be specified. The iteration process is divided into three stages, “Phase freedom iteration”, “Amplitude-Phase-Scale freedom iteration” and “Quantization”. During the phase freedom iteration only the phase values of the pixels in the signal window may develop freely. The “noise” outside the signal window is suppressed. This stage is normally used to increase the diffraction efficiency of the diffractive element. It may, however, destroy the band limitation of the phase distribution superimposed onto the intensity signal. Thus, in the case of speckle reduction no “Phase freedom iteration” should be performed. If a complex signal type is selected, this stage is disabled. The only constraint of this iterative Fourier transform algorithm in the space domain during the “Amplitude-Phase-Scale freedom iteration” is the replacement of the intensity in the signal window, that is, the freedom parameters are amplitude and phase outside the signal window and phase freedom in the signal window (only in the case of intensity signals). The constraints in the Fourier domain depend on whether “Soft” or “Hard” coding is selected in the “Coding” group. Hard coding forces the transmission function of every iteration to lie in the set of legal transmission functions defined by the “DOE Type” group. If soft coding was selected, the transmission function is slowly forced towards this set during the
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Figure 10 During the iteration the diffraction efficiency and SNR value of the current diffractive element distribution are appended to a log file displayed on the DIGIOPT desktop (left). These values may be pasted directly into standard windows software and displayed as graphs, tables, etc. (right). iteration stage. This leads only to small changes in the phase relation of neighbouring pixels after each iteration, thus speckles may be avoided. In our example, we performed 200 iterations with amplitude, phase and scale freedom. The last stage, “Quantization”, enables the quantization of the diffractive element. If a non-zero value is specified, a subdialog is displayed asking for the number of quantization levels and coding type. The coding type is either hard or soft as in the “Amplitude-Phase-Scale freedom iteration”. In our example, we decided to perform 200 iterations of soft 4-level quantization. The “Log File” group lets the user create a log file displayed on the DIGIOPT desktop during the iteration process. The log file enables the examination of the development of the transmission function of the diffractive element. Figure 10 shows a log file window and how this can be pasted into standard software and displayed as a graph. Figure 9 shows the results of the applied iterative Fourier transform algorithm. The upper left image is the continuous diffractive phase element that was created after the “Amplitude-Phase-Scale freedom iteration”. Its sinc-interpolated reconstruction is shown to the right. The lower row shows the 4-level quantized diffractive element obtained after the “Quantization” stage and its sinc-interpolated reconstruction. The reconstruction of the continuous element has a good signal-to-noise ratio and is practically speckle-free. Unfortunately, the computation of quantized non-periodic diffractive elements with few quantization levels is a very difficult task. Although the reconstruction of the 4-level quantized diffractive element does not contain any speckles, the signal-to-noise ratio is poor. This is easily seen in Figure 9.
4. Conclusion We have stated criteria for the goodness of software systems for the design and simulation of paraxial diffractive elements. The DIGIOPT system was described and demonstrated by generating a non-periodic diffractive phase element giving a continuous speckle-free reconstruction of a desired pattern which is a complicated task. The computation of the diffractive element was divided into three stages; (i) finding an object independent band-limited phase distribution, (ii) finding an initial object dependent band-limited phase distribution and (iii) generation of the diffractive element. Although not necessary, many low level operators were used Figure 9 instead of fewer high level ones to show the flexibility and functionality of DIGIOPT. The example demonstrated the ease of use of DIGIOPT even in solving non-trivial design problems.
The upper row shows the computed continuous DPE with its sinc-interpolated reconstruction. The 4-level quantized DPE is shown in the lower row.
The programming language of DIGIOPT was demonstrated by two small examples, one producing an animation sequence containing the diffraction pattern of an image at different distances behind the image and one creating a wavefront window from an analytical expression. The programming language offers many interesting features for the advanced user. Considering the non-trivial design example, the variety of features of DIGIOPT and the ease of use, DIGIOPT seems to be a system fulfilling the criteria stated for the goodness of software systems for the design and simulation of diffractive optical systems. Thus, DIGIOPT could possibly function as a link between research and industry in the field of optical signal processing based on paraxial diffractive elements.
5. References [1] D. C. O’Shea, J. W. Beltic, and Menelaous Poutous, “Binary-mask generation for diffractive optical elements using microcomputers,” Appl. Optics, Vol. 32, No. 14, May 1993. [2] F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. in Physics 54, 1481– 1571, 1991. [3] H. Aagedal, T. Beth, F. Reichel, H. Schwarzer and S. Teiwes, “Opto-electronical Hybrid System for Rapid Prototyping of Paraxial Diffractive Elements,” Presented at Spring Topical Meeting, Spatial Light Modulators and Applications, Opt.Soc.Am., Salt Lake City, 1995. [4] H. Aagedal, T. Beth, H. Schwarzer and S. Teiwes, “Modern concepts for computer-aided design in diffractive optics,” OSA Proceedings of the International Optical Design Conference, G.W. Forbes, ed., Vol. 22, 257–260, 1994. [5] R.W. Gerchberg and W.O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246, 1972. [6] M.S. Scivier and M.A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J.Opt.Soc.Am. A 2, 693–697, 1985. [7] R. Bräuer, F. Wyrowski and O. Bryngdahl, “Diffusors in digital holography,” J.Opt.Soc.Am. A 8, 572–578, 1991. [8] S. Teiwes, H. Schwarzer, H. Aagedal, M. Schmid. T. Beth, F. Reichel and F. Wyrowski, “Speckle-free Signal Reconstruction with Non-periodic Diffractive Elements on Liquid-crystal SLM,” Presented at Spring Topical Meeting, Spatial Light Modulators and Applications, Opt.Soc.Am., Salt Lake City, 1995.
